**Differentiation** is a method to find rates of change. It is an important topic for the JEE exam. Derivative of a function y = f(x) of a variable x is the rate of change of y, with respect to the rate of change of x. This article helps you to learn the derivative of a function, standard derivatives, theorems of derivatives, differentiation of implicit functions and higher order derivatives, along with solved examples.

**Related Topics:**

## How to Differentiate a Function

The differentiation of a function is a way to show the rate of change of a function at a given point. For real-valued functions, it is the slope of the tangent line at a point on a graph.

The derivative of y with respect to x is defined as the change in y over the change in x, as the distance between x_{0} and x_{1} becomes infinitely small (infinitesimal). The derivative is often written as dy/dx.

In mathematical terms, if f is a real-valued function, and a is a point in its domain of definition, the derivative of f at a is defined by

## Standard Derivatives

* (Chain rule or function of a function rule)*

4) **Derivatives of trigonometric functions**

5) **Derivatives of inverse trigonometric functions**

6) **Exponential and logarithmic functions**

7) **Hyperbolic functions**

8) **Inverse hyperbolic functions**

## Some Standard Substitution

### Expression Substitution

Simple tricks to solve complicated differential equations are listed below.

(i) If a function contains

(ii) If a function contains

(iii) If a function contains

## Theorems of Derivatives

Find some of the important theorem results below.

**Example 1: Find dy/dx for y = x sinx log x**

**Solution:**

**Example 2: Find dy/dx for y = sin(x ^{2} + 1).**

**Solution:**

= 2x cos (x^{2} + 1)

**Differentiation of Implicit Function**

Implicit differentiation, also known as the **chain rule**, differentiate both sides of an equation with two given variables by considering one of the variables as a function of the second variable. In short, differentiate the given function with respect to x and solve for dy/dx. Let us have a look at some examples.

**Example 1:** If x^{2} + 2xy + y^{3} = 4, find dy/dx.

**Solution:** Differentiating both sides w.r.t. x, we get

**Example 2:** Differentiate log sin x w.r.t

**Solution:** Let u = log sin x and v =

### Higher Order Derivatives

The differentiation process can be continued up to the nth derivation of a function. Usually, we deal with the first-order and second-order derivatives of the functions.

dy/dx is the first derivative of y w.r.t x.

d^{2}y/dx^{2} is the second derivative of y w.r.t x.

Similarly, finding the third, fourth, fifth and successive derivatives of any function, say g(x), which are known as higher-order derivatives of g(x). The

nth order derivative numerical notation is g^{n}(x) or d^{n}y/dx^{n}

**Example:** If

**Solution:**

= (1-2x)dy/dx (from eqn (i))

Hence proved.

## Video Lessons

## Methods of Differentiation – JEE Solved Questions

## Important Theorems of Differentiation for JEE

## Frequently Asked Questions

### What do you mean by differentiation in mathematics?

Differentiation is the process of finding the derivative of a function.

### Give the product rule of differentiation.

Product rule: (d/dx) (uv) = u (dv/dx) + v (du/dx).

### Give the quotient rule of differentiation.

Quotient rule: (d/dx)(u/v) = (v (du/dx) – u (dv/dx))/v^{2}.

### What is the derivative of cot x w.r.t. x?

The derivative of cot x w.r.t.x = -cosec^{2}x.

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