Derivative Examples

Consider a function which involves the concept of a slope of a line. The rate of change of the y-axis is dependent on the rate of change of the x-axis. Pick two points on the line such that the change in x [Î”x] and y [Î”y] defined as the slope is given by [Î”y] / [Î”x]. The concept of the derivative is derived from limits. The derivative examples and previous year JEE Main problems on derivatives are presented in this article.

Derivative Definition

The change in the output value of the function with respect to the change in the input value is defined as the derivative of a function of a real variable. The process involved in finding a derivative is called differentiation. It is a relationship that denotes the change of the dependent variable in terms of an independent variable represented by an equation. f ’(x) is the derivative of the function f (x).

Derivative Formula

The instantaneous rate of change of a function can be termed as derivative. A few derivative examples can also be seen here. It is represented as follows:

Derivatives of some basic functions:

Trigonometric and Inverse Trigonometric Functions

Previous Year JEE main Problems On Derivatives

The problems that appeared in the previous year JEE main examination are discussed below. Solving Previous Year JEE main Problems On Derivatives will help the candidates to score high marks in the examination. It enables the students to understand the question paper pattern and weightage of marks for each topic.

Example 1: What is the derivative of (log tanx)?

Solution:

Let y = (log tanx)
Differentiating w.r.t. x, we get
[dy / dx] = (1 / tanx) sec2 x
= cosx / [cos2 x sinx]
= (2 / 2) (1 / cosx sinx)
=  2 cosec 2x

Example 2:  If f(x) = logx (logx), then find f′(x) at x = e.

Solution:

f(x) = logx (logx)
= log (log x) / logx
f′(x)= 1 / x − 1 / x log (logx) / (logx)2
⇒ f′(e) = [1 / e − 0] / 1
= 1 / e

Example 2: What is the derivative of the function x2exxinx?

Solution:

Let f(x) = x2exxinx
⇒f′(x) = x2 (d / dx) [ex sinx] + [ex sinx] (d / dx) (x2)
= xex (2sinx + xsinx + xcosx)

Example 3:  If xmyn = (x + y)m+n, then find (dy / dx).

Solution:

Example 4: If √1 - x2 + √1 - y2 = a (x - y), then dy / dx is

Solution:

Example 5: Let y be an implicit function of x defined by x2x − 2xx coty − 1 = 0. Then y'(1) equals

Solution: