Differentiation Formulas

Differentiation is the mathematical process of determining the finding of a function, which represents the rate at which the function’s value changes with respect to its independent variable. The derivative, denoted as \frac{d}{dx}f(x), provides a precise measure of the function’s instantaneous rate of change.

This operation forms the basis of differential calculus, with specific formulas and rules applicable to algebraic, trigonometric, exponential, logarithmic, and inverse trigonometric functions.

The derivative of 𝑓(x) at x is defined as the limit as h approaches 0:

\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x + h) - f(x)}{(x + h) - x}

Mathematically,

\frac{dy}{dx} = f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

This limit represents the instantaneous rate of change of y with respect to x or the slope of the tangent line to the curve y = 𝑓(x) at the point (x, 𝑓(x)).

Differentiation formulas are used to find the differentiation of the various functions. The first principal formula states that, for any function 𝑓(x) its derivative with respect to x is,

f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

Basic Differentiation Formulas

The differentiation formulas for some elementary functions are:

Differentiation of Trigonometric Functions

Derivatives of the trigonometric functions are:

Differentiation of Inverse Trigonometric Functions

The differentiation formulas for the Inverse trigonometric functions are:

Differentiation of Hyperbolic Functions

Let's discuss the Differentials of Hyperbolic functions.

Differentiation Rules

Various rules of finding the derivative of functions have been given below:

Differentiation of Special Functions

If we have two parametric functions x = 𝑓(t), y = g(t), where t is the parameter, then the differentiation of parametric functions is as follows,

As dy/dt = g'(t) and dx/dt = 𝑓'(t) then dy/dx is given by:

\frac{dy}{dx} = \frac{\,dy/dt\,}{\,dx/dt\,} = \frac{g'(t)}{f'(t)}

Implicit Differentiation

If y is related to x but can not conveniently expressed in the form y = 𝑓(x) but can be expressed in the form 𝑓(x,y) = 0, then we say that y is an implicit function of x. In the case of implicit function dy/dx can be found by following steps.

(a) Differentiate each term of 𝑓(x, y) = 0 with respect to x.

(b) Collect the terms containing dy/dx on one side and the terms not involving dy/dx on the other side.

(c) Express dy/dx as a function of x or y or both.

Example: Find the differentiation of x² + y² + 4xy = 0

Solution:

x² + y² + 4xy = 0

Differentiating with respect to x,

2x + 2ydy/dx + 4(xdy/dx + y) = 0
⇒ 2x + 4y + 2dy/dx(y + 2x) = 0
⇒ x + 2y + dy/dx(y + 2x) = 0
⇒ dy/dx(y + 2x) = -(x + 2y)
⇒ dy/dx = -(x + 2y)/(y + 2x)

Higher Order Differentiation

Higher order differentiation is nothing, but the differentiation of a function more than one time suppose we have a function y = 𝑓(x) then its differential in higher order is calculated as,

First Derivative = \frac{dy}{dx} = f'(x)

Second Derivative = \frac{d^2 y}{dx^2} = f''(x)

Third Derivative = \frac{d^3 y}{dx^3} = f'''(x)

....
....
....

n^th Derivative =\frac{d^n y}{dx^n} = f^{(n)}(x)

This can be understood using the example added below,

Example: Find the second-order derivative of 𝑓(x) = 4x⁴ + 3x³ + 2x² + x + 1

Solution:

𝑓(x) = 4x⁴ + 3x³ + 2x² + x + 1

Differentiating with respect to x,

𝑓^'(x) = 4(4x³) + 3(3x²) + 2(2x) + 1 + 0
⇒ 𝑓'(x) = 16x³ + 9x² + 4x + 1

For second-order derivative differentiating with respect to x,

𝑓''(x) = 16(3x²) + 9(2x) + 4 + 0
⇒ 𝑓''(x) = 48x² + 18x + 4

This is the required second-order derivative.

Solved Examples of Differentiation Formulas

Example 1: Find the differentiation of y = 4x³ + 7x² + 11x + 12

Solution:

Given, y = 4x³ + 7x² + 11x + 12

Differentiating with respect to x,

dy/dx = 4(3x²) + 7(2x) + 11(1) + 0

⇒ dy/dx = 12x² + 14x + 11

This is the required differentiation

Example 2: Find the differentiation of y = cos(log x)

Solution:

Given, y = cos(log x)

Differentiating with respect to x,

dy/dx = d/dx{cos (log x)}

⇒ dy/dx = -sin (log x).{d/dx(log x)}
⇒ dy/dx = -sin (log x).(1/x)

This is the required differentiation

Practice Problems on Differentiation Formulas

Problem 1: Find the derivative of the function f(x) = 3x² + 5x - 2.

Problem 2: Determine the derivative of g(x) = 1/x.

Problem 3: Find the derivative of h(x) = \sqrt{x^3 + 2x - 1}.

Problem 4: Determine the derivative of y(x) = e^2x.

Problem 5: Find the derivative of f(x) = \ln(x^2 + 3x).