What is a derivative in calculus?

In calculus, a derivative measures how a function changes as its input changes. It represents the instantaneous rate of change and gives the slope of the function's tangent line at a specific point.

How do you interpret the derivative of a function?

The derivative of a function at a specific point gives the slope of the tangent line to the function's graph at that point. This slope indicates how steeply the function is increasing or decreasing: A positive derivative indicates the function is increasing. A negative derivative indicates the function is decreasing. A zero derivative indicates a local maximum, minimum, or a horizontal inflection point.

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Derivative Function Calculus

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Delve into Derivatives

Derivative represents the rate of change of a function with respect to a variable. If we consider a curve...

Learn The Various Methods of Differentiation

Differentiation is a fundamental concept in calculus that deals with how things change. Specifically,...

Limits and Derivatives

Limits and Derivatives are fundamental concepts in calculus that lay the groundwork for more advanced topics.

First Derivative Test

The First Derivative Test is a crucial concept in calculus used to determine the behavior of a function, particularly in identifying local maxima, minima, and points of inflection.

Explore The Types of Matrices

A matrix is actually an arrangement of numbers, symbols, or expressions in rows and columns.

Understand The Different Operations On Matrices

Addition, subtraction and multiplication are the elementary operations on the matrix. In order to add...

Determinant of a Matrix

The determinant of a matrix is an important number in math that tells us if a matrix can be inverted.

Derivative Function Calculus

In mathematics calculus, the derivative of a function at a particular point provides the slope of the tangent line to the function's graph at that point. This slope represents the rate of change of the function with respect to its input variable. If we denote a function by f(x), its derivative is often represented as $f^{'} (x) or \frac{df}{d x}$

1.0Introduction to Function Derivatives

What is a Derivative?

At its simplest, the derivative of a function at a given point is the slope of the tangent line to the function's graph at that point. This slope indicates how steeply the function's value is changing at that specific input value. Mathematically, for a function f(x), the derivative at x = a is defined as:

$f^{'} (a) = lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h}$

This limit, if it exists, captures the instantaneous rate of change of the function.

2.0Derivative of a Function

The derivative of a function f(x) at x = a is defined as the limit of the function's average rate of change as the interval around a approaches zero. Mathematically, this is expressed as:

$f^{'} (a) = lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h}$

If this limit exists, f'(a) represents the instantaneous rate of change of the function at x = a. The process of finding the derivative is called differentiation.

Notation

Several notations are commonly used to denote the derivative of a function:

f'(x) (Leibniz notation)
$\frac{df}{dx}$ (Leibniz notation)
Df(x) (Euler notation)

For higher-order derivatives, which represent the derivative of the derivative, the notation extends to f''(x) for the second derivative, f'''(x) for the third derivative, and so on.

3.0Derivative Function Calculus Rules

Basic Derivative Function Formulas are

Constant Function

$\frac{d}{dx} [c] = 0 Wh erec i s a co n s t an t .$

Power Rule

$\frac{d}{d x} [x^{n}] = n x^{n - 1}$ Applicable for any real number n.

Constant Multiple Rule

$\frac{d}{dx} [c \times f (x)] = c \times f^{'} (x)$ , where c is a constant.

Sum Rule

$\frac{d}{dx} [f (x) + g (x)] = f^{'} (x) + g^{'} (x)$

Difference Rule

$\frac{d}{d x} [f (x) - g (x)] = f^{'} (x) - g^{'} (x)$

Product Rule

$\frac{d}{d x} [f (x) \times g (x)] = f^{'} (x) g (x) + f (x) g^{'} (x)$

If 3 functions are involved

$\frac{d}{d x} [f (x) \times g (x) \times h (x)] = f^{'} (x) g (x) h (x) + f (x) g^{'} (x) h (x) + f (x) g (x) h^{'} (x)$

Quotient Rule

$\frac{d}{d x} [\frac{f ( x )}{g ( x )}] = \frac{f ^{'} ( x ) g ( x ) - f ( x ) g ^{'} ( x )}{g ( x ) ^{2}}$

Chain Rule

$\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \times g^{'} (x)$

4.0Derivatives of Standard Functions

Exponential Functions

$\frac{d}{d x} [e^{x}] = e^{x}$
$\frac{d}{dx} [a^{x}] = a^{x} ln (a)$

Where a is a constant.

Logarithmic Functions

$\frac{d}{d x} [ln (x)] = \frac{1}{x}$
$\frac{d}{d x} [lo g_{a} (x)] = \frac{1}{x l n ( a )}$

Trigonometric Functions

$\frac{d}{d x} [sin (x)] = cos (x)$
$\frac{d}{d x} [cos (x)] = - sin (x)$
$\frac{d}{d x} [tan (x)] = sec^{2} (x)$
$\frac{d}{d x} [cosec (x)] = - cosec (x) cot (x)$
$\frac{d}{d x} [sec (x)] = sec (x) tan (x)$
$\frac{d}{d x} [cot (x)] = - cosec^{2} (x)$

Inverse Trigonometric Functions

$\frac{d}{d x} [arcsin (x)] = \frac{1}{1 - x ^{2}}$
$\frac{d}{d x} [arccos (x)] = - \frac{1}{1 - x ^{2}}$
$\frac{d}{d x} [arctan (x)] = \frac{1}{1 + x ^{2}}$
$\frac{d}{d x} [arccot (x)] = - \frac{1}{1 + x ^{2}}$
$\frac{d}{d x} [arcsec (x)] = \frac{1}{∣ x ∣ x ^{2} - 1}$
$\frac{d}{d x} [arccosec (x)] = - \frac{1}{∣ x ∣ x ^{2} - 1}$

Implicit Differentiation

For functions not explicitly solved for one variable in terms of another, implicit differentiation is used:

Implicit Differentiation

$If F (x, y) = 0, then \frac{d F}{d x} = F_{x}^{'} + F_{y}^{'} \cdot \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{F _{x}^{'}}{F _{y}^{'}}$

To find $\frac{d y}{d x}$ of implicit curves, we differentiate each term with respect to x regarding y as a function of x and then collect term with \frac{d y}{d x} together on one side

Differentiation of homogeneous relation

$\frac{d y}{d x} = \frac{y}{x} \frac{d ^{2} y}{d x ^{2}} = 0$

5.0Higher-Order Derivatives

Second Derivative (f''(x))

$f^{''} (x) = \frac{d}{d x} [f^{'} (x)]$

The second derivative represents the rate of change of the first derivative, providing information about the concavity of the function.

n^th Derivative (fⁿ(x))

$f^{(n)} (x) = \frac{d ^{n}}{d x ^{n}} [f (x)]$

Higher-order derivatives can provide deeper insights into the behavior of functions, such as inflection points and more detailed curvature information.

6.0Derivative Function Calculus Examples

Differentiate $f (x) = x^{4}$

Solution: Using the power rule $\frac{d}{d x} [x^{n}] = n x^{n - 1}$ , $f^{'} (x) = 4 x^{3}$

Differentiate $g (x) = x^{3} - 2 x^{2} + x - 5$

Solution: Using the sum and difference rule,

$g^{'} (x) = \frac{d}{dx} [x^{3}] - \frac{d}{dx} [2 x^{2}] + \frac{d}{dx} [x] - \frac{d}{dx} [5]$

$g^{'} (x) = 3 x^{2} - 4 x + 1$

Differentiate $h (x) = x^{2} sin (x)$

Solution: Using the product rule $\frac{d}{d x} [u (x) v (x)] = u^{'} (x) v (x) + u (x) v^{'} (x)$ ,

Let u(x) = x² and v(x) = sin(x),

u'(x) = 2x and v'(x) = cos(x)

Therefore,

h'(x) = (2x) sin(x) + (x²) cos(x)

h'(x) = 2xsin(x) + x²cos(x)

Differentiate $k (x) = \frac{x ^{2} + 1}{x}$

Solution: Using the quotient rule $\frac{d}{d x} [\frac{u ( x )}{v ( x )}] = \frac{u ^{'} ( x ) v ( x ) - u ( x ) v ^{'} ( x )}{v ( x ) ^{2}}$ ,

Let u(x) = x² + 1 and v(x) = x,

u'(x) = 2x and v'(x) = 1

Therefore,

$k^{'} (x) = \frac{( 2 x ) ( x ) - ( x ^{2} + 1 ) ( 1 )}{x ^{2}} k^{'} (x) = \frac{2 x ^{2} - x ^{2} - 1}{x ^{2}}$

$k^{'} (x) = \frac{x ^{2} - 1}{x ^{2}}$

$k^{'} (x) = 1 - \frac{1}{x ^{2}}$

Differentiate m(x) = sin(x²)

Solution: Using the chain rule $\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$ ,

Let u = x2, so m(x) = sin(u),

$\frac{d}{du} sin (u) = cos (u)$

$\frac{d}{d x} (x^{2}) = 2 x$

Therefore, m'(x) = 2x cos(x2)

Differentiate p(x) = e^3x

Solution: Using the exponential rule $\frac{d}{d x} [e^{u (x)}] = e^{u (x)} \cdot u^{'} (x)$ ,

Let u(x) = 3x,

Then u'(x) = 3

Therefore,

p'(x) = e^3x ·3

p'(x) = 3e^3x

Differentiate q(x) = ln(x² + 1)

Solution: Using the chain rule and the logarithm rule $\frac{d}{d x} [ln (u (x))] = \frac{1}{u ( x )} \cdot u^{'} (x)$ ,

Let u(x) = x² + 1,

Then u'(x) = 2x

Therefore, $q^{'} (x) = \frac{1}{x ^{2} + 1} \cdot 2 x$

$q^{'} (x) = \frac{2 x}{x ^{2} + 1}$

Differentiate x² + y² = 25

Solution: To find $\frac{dy}{dx}$ , differentiate both sides with respect to x,

$\frac{d}{dx} [x^{2}] + \frac{d}{dx} [y^{2}] = \frac{d}{dx} [25]$

$2 x + 2 y \frac{d y}{d x} = 0$

Solving for $\frac{d y}{d x}$ ,

$2 y \frac{d y}{d x} = - 2 x$

$\frac{d y}{d x} = - \frac{x}{y}$

Differentiate r(x) = x³ for higher order up to 3

Solution: First Derivative:

r'(x) = 3x²

Second Derivative:

$r^{''} (x) = \frac{d}{d x} [3 x^{2}] = 6 x$

Third Derivative:

$r^{'''} (x) = \frac{d}{d x} [6 x] = 6$

7.0Derivative Function Calculus Practice Question

Find the derivative of the function f(x) = 5x⁴.
Determine the derivative of g(x) = 3x³ – 4x² + 7x – 2.
Compute the derivative of the function h(x) = x²cos(x).
Find the derivative of k(x)= $\frac{x ^{2} + 1}{x}$
Determine the derivative of the function m(x) = sin(3x²).
Find $\frac{d y}{d x}$
if x² + y² = 16.
Calculate the derivative of p(x) = e^4x.
Find the derivative of q(x) = ln (x² + 3).
Compute the derivative of r(x) = tan(x).
Determine the second derivative of s(x) = x³ – 3x² + 4x – 5.

8.0Solved Questions on Derivative Function Calculus

Q. What are the basic rules of differentiation?

Ans: The basic rules of differentiation include:

Constant Rule: $\frac{d}{dx} [c] = 0$
Power Rule: $\frac{d}{dx} [x^{n}] = nx^{n - 1}$
Constant Multiple Rule: $\frac{d}{d x} [c \cdot f (x)] = c \cdot f^{'} (x)$
Sum Rule: $\frac{d}{d x} [f (x) + g (x)] = f^{'} (x) + g^{'} (x)$
Difference Rule: $\frac{d}{d x} [f (x) - g (x)] = f^{'} (x) - g^{'} (x)$
Product Rule: $\frac{d}{d x} [f (x) g (x)] = f^{'} (x) g (x) + f (x) g^{'} (x)$
Quotient Rule: $\frac{d}{d x} [\frac{f ( x )}{g ( x )}] = \frac{f ^{'} ( x ) g ( x ) - f ( x ) g ^{'} ( x )}{g ( x ) ^{2}}$
Chain Rule: $\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$

Q. How do you find the derivative of a trigonometric function?

Ans: The derivatives of the basic trigonometric functions are:

$\frac{d}{d x} [sin (x)] = cos (x)$
$\frac{d}{d x} [cos (x)] = - sin (x)$
$\frac{d}{d x} [tan (x)] = sec^{2} (x)$
$\frac{d}{d x} [cosec (x)] = - cosec (x) cot (x)$
$\frac{d}{d x} [sec (x)] = sec (x) tan (x)$
$\frac{d}{d x} [cot (x)] = - cosec^{2} (x)$

Q. What is implicit differentiation?

Ans: Implicit differentiation is a technique used to find the derivative of a function that is not explicitly solved for one variable in terms of another. For example, if you have an equation involving both x and y, such as x² + y² = 1, you differentiate both sides with respect to x and solve for $\frac{d y}{d x} .$

Q. How do you differentiate exponential and logarithmic functions?

Ans: For exponential functions:

$\frac{d}{d x} [e^{x}] = e^{x}$
$\frac{d}{dx} [a^{x}] = a^{x} ln (a)$

For logarithmic functions:

$\frac{d}{dx} [ln (x)] = \frac{1}{x}$
$\frac{d}{d x} [lo g_{a} (x)] = \frac{1}{x l n ( a )}$

Q. How do you handle derivatives of inverse trigonometric functions?

Ans: The derivatives of the basic inverse trigonometric functions are:

$\frac{d}{d x} [arcsin (x)] = \frac{1}{1 - x ^{2}}$
$\frac{d}{d x} [arccos (x)] = - \frac{1}{1 - x ^{2}}$
$\frac{d}{dx} [arctan (x)] = \frac{1}{1 + x ^{2}}$
$\frac{d}{dx} [arccot (x)] = - \frac{1}{1 + x ^{2}}$
$\frac{d}{d x} [arcsec (x)] = \frac{1}{∣ x ∣ x ^{2} - 1}$
$\frac{d}{d x} [arccsc (x)] = - \frac{1}{∣ x ∣ x ^{2} - 1}$

Join ALLEN!

(Session 2026 - 27)

Name

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Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Delve into Derivatives

Derivative represents the rate of change of a function with respect to a variable. If we consider a curve...

Learn The Various Methods of Differentiation

Differentiation is a fundamental concept in calculus that deals with how things change. Specifically,...

Limits and Derivatives

Limits and Derivatives are fundamental concepts in calculus that lay the groundwork for more advanced topics.

First Derivative Test

The First Derivative Test is a crucial concept in calculus used to determine the behavior of a function, particularly in identifying local maxima, minima, and points of inflection.

Explore The Types of Matrices

A matrix is actually an arrangement of numbers, symbols, or expressions in rows and columns.

Understand The Different Operations On Matrices

Addition, subtraction and multiplication are the elementary operations on the matrix. In order to add...

Determinant of a Matrix

The determinant of a matrix is an important number in math that tells us if a matrix can be inverted.

Derivative Function Calculus

In mathematics calculus, the derivative of a function at a particular point provides the slope of the tangent line to the function's graph at that point. This slope represents the rate of change of the function with respect to its input variable. If we denote a function by f(x), its derivative is often represented as $f^{'} (x) or \frac{df}{d x}$

1.0Introduction to Function Derivatives

What is a Derivative?

At its simplest, the derivative of a function at a given point is the slope of the tangent line to the function's graph at that point. This slope indicates how steeply the function's value is changing at that specific input value. Mathematically, for a function f(x), the derivative at x = a is defined as:

$f^{'} (a) = lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h}$

This limit, if it exists, captures the instantaneous rate of change of the function.

2.0Derivative of a Function

The derivative of a function f(x) at x = a is defined as the limit of the function's average rate of change as the interval around a approaches zero. Mathematically, this is expressed as:

$f^{'} (a) = lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h}$

If this limit exists, f'(a) represents the instantaneous rate of change of the function at x = a. The process of finding the derivative is called differentiation.

Notation

Several notations are commonly used to denote the derivative of a function:

f'(x) (Leibniz notation)
$\frac{df}{dx}$ (Leibniz notation)
Df(x) (Euler notation)

For higher-order derivatives, which represent the derivative of the derivative, the notation extends to f''(x) for the second derivative, f'''(x) for the third derivative, and so on.

3.0Derivative Function Calculus Rules

Basic Derivative Function Formulas are

Constant Function

$\frac{d}{dx} [c] = 0 Wh erec i s a co n s t an t .$

Power Rule

$\frac{d}{d x} [x^{n}] = n x^{n - 1}$ Applicable for any real number n.

Constant Multiple Rule

$\frac{d}{dx} [c \times f (x)] = c \times f^{'} (x)$ , where c is a constant.

Sum Rule

$\frac{d}{dx} [f (x) + g (x)] = f^{'} (x) + g^{'} (x)$

Difference Rule

$\frac{d}{d x} [f (x) - g (x)] = f^{'} (x) - g^{'} (x)$

Product Rule

$\frac{d}{d x} [f (x) \times g (x)] = f^{'} (x) g (x) + f (x) g^{'} (x)$

If 3 functions are involved

$\frac{d}{d x} [f (x) \times g (x) \times h (x)] = f^{'} (x) g (x) h (x) + f (x) g^{'} (x) h (x) + f (x) g (x) h^{'} (x)$

Quotient Rule

$\frac{d}{d x} [\frac{f ( x )}{g ( x )}] = \frac{f ^{'} ( x ) g ( x ) - f ( x ) g ^{'} ( x )}{g ( x ) ^{2}}$

Chain Rule

$\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \times g^{'} (x)$

4.0Derivatives of Standard Functions

Exponential Functions

$\frac{d}{d x} [e^{x}] = e^{x}$
$\frac{d}{dx} [a^{x}] = a^{x} ln (a)$

Where a is a constant.

Logarithmic Functions

$\frac{d}{d x} [ln (x)] = \frac{1}{x}$
$\frac{d}{d x} [lo g_{a} (x)] = \frac{1}{x l n ( a )}$

Trigonometric Functions

$\frac{d}{d x} [sin (x)] = cos (x)$
$\frac{d}{d x} [cos (x)] = - sin (x)$
$\frac{d}{d x} [tan (x)] = sec^{2} (x)$
$\frac{d}{d x} [cosec (x)] = - cosec (x) cot (x)$
$\frac{d}{d x} [sec (x)] = sec (x) tan (x)$
$\frac{d}{d x} [cot (x)] = - cosec^{2} (x)$

Inverse Trigonometric Functions

$\frac{d}{d x} [arcsin (x)] = \frac{1}{1 - x ^{2}}$
$\frac{d}{d x} [arccos (x)] = - \frac{1}{1 - x ^{2}}$
$\frac{d}{d x} [arctan (x)] = \frac{1}{1 + x ^{2}}$
$\frac{d}{d x} [arccot (x)] = - \frac{1}{1 + x ^{2}}$
$\frac{d}{d x} [arcsec (x)] = \frac{1}{∣ x ∣ x ^{2} - 1}$
$\frac{d}{d x} [arccosec (x)] = - \frac{1}{∣ x ∣ x ^{2} - 1}$

Implicit Differentiation

For functions not explicitly solved for one variable in terms of another, implicit differentiation is used:

Implicit Differentiation

$If F (x, y) = 0, then \frac{d F}{d x} = F_{x}^{'} + F_{y}^{'} \cdot \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{F _{x}^{'}}{F _{y}^{'}}$

To find $\frac{d y}{d x}$ of implicit curves, we differentiate each term with respect to x regarding y as a function of x and then collect term with \frac{d y}{d x} together on one side

Differentiation of homogeneous relation

$\frac{d y}{d x} = \frac{y}{x} \frac{d ^{2} y}{d x ^{2}} = 0$

5.0Higher-Order Derivatives

Second Derivative (f''(x))

$f^{''} (x) = \frac{d}{d x} [f^{'} (x)]$

The second derivative represents the rate of change of the first derivative, providing information about the concavity of the function.

n^th Derivative (fⁿ(x))

$f^{(n)} (x) = \frac{d ^{n}}{d x ^{n}} [f (x)]$

Higher-order derivatives can provide deeper insights into the behavior of functions, such as inflection points and more detailed curvature information.

6.0Derivative Function Calculus Examples

Differentiate $f (x) = x^{4}$

Solution: Using the power rule $\frac{d}{d x} [x^{n}] = n x^{n - 1}$ , $f^{'} (x) = 4 x^{3}$

Differentiate $g (x) = x^{3} - 2 x^{2} + x - 5$

Solution: Using the sum and difference rule,

$g^{'} (x) = \frac{d}{dx} [x^{3}] - \frac{d}{dx} [2 x^{2}] + \frac{d}{dx} [x] - \frac{d}{dx} [5]$

$g^{'} (x) = 3 x^{2} - 4 x + 1$

Differentiate $h (x) = x^{2} sin (x)$

Solution: Using the product rule $\frac{d}{d x} [u (x) v (x)] = u^{'} (x) v (x) + u (x) v^{'} (x)$ ,

Let u(x) = x² and v(x) = sin(x),

u'(x) = 2x and v'(x) = cos(x)

Therefore,

h'(x) = (2x) sin(x) + (x²) cos(x)

h'(x) = 2xsin(x) + x²cos(x)

Differentiate $k (x) = \frac{x ^{2} + 1}{x}$

Solution: Using the quotient rule $\frac{d}{d x} [\frac{u ( x )}{v ( x )}] = \frac{u ^{'} ( x ) v ( x ) - u ( x ) v ^{'} ( x )}{v ( x ) ^{2}}$ ,

Let u(x) = x² + 1 and v(x) = x,

u'(x) = 2x and v'(x) = 1

Therefore,

$k^{'} (x) = \frac{( 2 x ) ( x ) - ( x ^{2} + 1 ) ( 1 )}{x ^{2}} k^{'} (x) = \frac{2 x ^{2} - x ^{2} - 1}{x ^{2}}$

$k^{'} (x) = \frac{x ^{2} - 1}{x ^{2}}$

$k^{'} (x) = 1 - \frac{1}{x ^{2}}$

Differentiate m(x) = sin(x²)

Solution: Using the chain rule $\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$ ,

Let u = x2, so m(x) = sin(u),

$\frac{d}{du} sin (u) = cos (u)$

$\frac{d}{d x} (x^{2}) = 2 x$

Therefore, m'(x) = 2x cos(x2)

Differentiate p(x) = e^3x

Solution: Using the exponential rule $\frac{d}{d x} [e^{u (x)}] = e^{u (x)} \cdot u^{'} (x)$ ,

Let u(x) = 3x,

Then u'(x) = 3

Therefore,

p'(x) = e^3x ·3

p'(x) = 3e^3x

Differentiate q(x) = ln(x² + 1)

Solution: Using the chain rule and the logarithm rule $\frac{d}{d x} [ln (u (x))] = \frac{1}{u ( x )} \cdot u^{'} (x)$ ,

Let u(x) = x² + 1,

Then u'(x) = 2x

Therefore, $q^{'} (x) = \frac{1}{x ^{2} + 1} \cdot 2 x$

$q^{'} (x) = \frac{2 x}{x ^{2} + 1}$

Differentiate x² + y² = 25

Solution: To find $\frac{dy}{dx}$ , differentiate both sides with respect to x,

$\frac{d}{dx} [x^{2}] + \frac{d}{dx} [y^{2}] = \frac{d}{dx} [25]$

$2 x + 2 y \frac{d y}{d x} = 0$

Solving for $\frac{d y}{d x}$ ,

$2 y \frac{d y}{d x} = - 2 x$

$\frac{d y}{d x} = - \frac{x}{y}$

Differentiate r(x) = x³ for higher order up to 3

Solution: First Derivative:

r'(x) = 3x²

Second Derivative:

$r^{''} (x) = \frac{d}{d x} [3 x^{2}] = 6 x$

Third Derivative:

$r^{'''} (x) = \frac{d}{d x} [6 x] = 6$

7.0Derivative Function Calculus Practice Question

Find the derivative of the function f(x) = 5x⁴.
Determine the derivative of g(x) = 3x³ – 4x² + 7x – 2.
Compute the derivative of the function h(x) = x²cos(x).
Find the derivative of k(x)= $\frac{x ^{2} + 1}{x}$
Determine the derivative of the function m(x) = sin(3x²).
Find $\frac{d y}{d x}$
if x² + y² = 16.
Calculate the derivative of p(x) = e^4x.
Find the derivative of q(x) = ln (x² + 3).
Compute the derivative of r(x) = tan(x).
Determine the second derivative of s(x) = x³ – 3x² + 4x – 5.

8.0Solved Questions on Derivative Function Calculus

Q. What are the basic rules of differentiation?

Ans: The basic rules of differentiation include:

Constant Rule: $\frac{d}{dx} [c] = 0$
Power Rule: $\frac{d}{dx} [x^{n}] = nx^{n - 1}$
Constant Multiple Rule: $\frac{d}{d x} [c \cdot f (x)] = c \cdot f^{'} (x)$
Sum Rule: $\frac{d}{d x} [f (x) + g (x)] = f^{'} (x) + g^{'} (x)$
Difference Rule: $\frac{d}{d x} [f (x) - g (x)] = f^{'} (x) - g^{'} (x)$
Product Rule: $\frac{d}{d x} [f (x) g (x)] = f^{'} (x) g (x) + f (x) g^{'} (x)$
Quotient Rule: $\frac{d}{d x} [\frac{f ( x )}{g ( x )}] = \frac{f ^{'} ( x ) g ( x ) - f ( x ) g ^{'} ( x )}{g ( x ) ^{2}}$
Chain Rule: $\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$

Q. How do you find the derivative of a trigonometric function?

Ans: The derivatives of the basic trigonometric functions are:

$\frac{d}{d x} [sin (x)] = cos (x)$
$\frac{d}{d x} [cos (x)] = - sin (x)$
$\frac{d}{d x} [tan (x)] = sec^{2} (x)$
$\frac{d}{d x} [cosec (x)] = - cosec (x) cot (x)$
$\frac{d}{d x} [sec (x)] = sec (x) tan (x)$
$\frac{d}{d x} [cot (x)] = - cosec^{2} (x)$

Q. What is implicit differentiation?

Ans: Implicit differentiation is a technique used to find the derivative of a function that is not explicitly solved for one variable in terms of another. For example, if you have an equation involving both x and y, such as x² + y² = 1, you differentiate both sides with respect to x and solve for $\frac{d y}{d x} .$

Q. How do you differentiate exponential and logarithmic functions?

Ans: For exponential functions:

$\frac{d}{d x} [e^{x}] = e^{x}$
$\frac{d}{dx} [a^{x}] = a^{x} ln (a)$

For logarithmic functions:

$\frac{d}{dx} [ln (x)] = \frac{1}{x}$
$\frac{d}{d x} [lo g_{a} (x)] = \frac{1}{x l n ( a )}$

Q. How do you handle derivatives of inverse trigonometric functions?

Ans: The derivatives of the basic inverse trigonometric functions are:

$\frac{d}{d x} [arcsin (x)] = \frac{1}{1 - x ^{2}}$
$\frac{d}{d x} [arccos (x)] = - \frac{1}{1 - x ^{2}}$
$\frac{d}{dx} [arctan (x)] = \frac{1}{1 + x ^{2}}$
$\frac{d}{dx} [arccot (x)] = - \frac{1}{1 + x ^{2}}$
$\frac{d}{d x} [arcsec (x)] = \frac{1}{∣ x ∣ x ^{2} - 1}$
$\frac{d}{d x} [arccsc (x)] = - \frac{1}{∣ x ∣ x ^{2} - 1}$

Frequently Asked Questions

What is a derivative in calculus?

How do you interpret the derivative of a function?

Frequently Asked Questions

What is a derivative in calculus?

How do you interpret the derivative of a function?

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1.0Introduction to Function Derivatives

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Constant Function

Power Rule

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Sum Rule

Difference Rule

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Quotient Rule

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Exponential Functions

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5.0Higher-Order Derivatives

Second Derivative (f''(x))

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Join ALLEN!

Related Articles:-

Delve into Derivatives

Learn The Various Methods of Differentiation

Limits and Derivatives

First Derivative Test

Explore The Types of Matrices

Understand The Different Operations On Matrices

Determinant of a Matrix

Derivative Function Calculus

1.0Introduction to Function Derivatives

What is a Derivative?

2.0Derivative of a Function

Notation

3.0Derivative Function Calculus Rules

Constant Function

Power Rule

Constant Multiple Rule

Sum Rule

Difference Rule

Product Rule

Quotient Rule

Chain Rule

4.0Derivatives of Standard Functions

Exponential Functions

Logarithmic Functions

Trigonometric Functions

Inverse Trigonometric Functions

Implicit Differentiation

5.0Higher-Order Derivatives

Second Derivative (f''(x))

nth Derivative (fn(x))

6.0Derivative Function Calculus Examples

7.0Derivative Function Calculus Practice Question

8.0Solved Questions on Derivative Function Calculus

Table of Contents

n^th Derivative (fⁿ(x))

n^th Derivative (fⁿ(x))