**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:**