Finding the derivative of a function is a method to find the rates of change. As far as the JEE Main exam is concerned, the derivative is a topic with great importance. Students can expect 2-3 questions from derivatives. Those who are preparing for the JEE Main should definitely gain knowledge of this topic and practise the previous years’ questions asked for the JEE Main. In this article, we will discuss the derivative of a function, important formulas of derivatives, higher-order derivatives, the derivative of a matrix, etc.
Let y = f(x) is a function. Then the derivative of y is denoted by dy/dx. So we can write dy/dx = f’(x). Finding the derivative of a function is also termed as differentiation. Derivative of a function of a variable x can be defined as the rate of change of y with respect to the rate of change of x. The derivative of a real-valued function is the slope of the tangent line at a point on a graph. Let us have a look at the standard derivatives.
Standard Formulas of Differentiation
1. (d/dx)xn = nxn-1
2. If c is a constant, (d/dx)c = 0
3. (d/dx) sin x = cos x
4. (d/dx) cos x = -sin x
5. (d/dx) tan x = sec2 x
6. (d/dx) sec x = sec x tan x
7. (d/dx) cosec x = -cosec x cot x
8. (d/dx) cot x = -cosec2x
9. (d/dx) ex = ex
10. (d/dx) log x = 1/x
Product Rule and Quotient Rule
The product rule is an important rule in differentiation. If u and v are two functions, then
Product rule id given by (d/dx) uv = u.(d/dx)v + v.(d/dx)u.
The quotient rule is given by (d/dx) u/v = [v. (d/dx) u – u.(d/dx) v]/v2
Let us discuss an example of product rule and quotient rule.
Example 1: Let y = x sin x. Find dy/dx.
Solution: Given that y = x sin x
We have two functions.
Let u = x and v = sin x
(d/dx) uv = u.(d/dx)v + v.(d/dx)u.
So dy/dx = x cos x + sin x
Example 2: Find f’(x) if f(x) = x2/(x+1).
Solution:
Given f(x) = x2/(x+1)
Let u = x2 and v = x+1
The quotient rule is given by (d/dx) u/v = [v. (d/dx) u – u.(d/dx) v]/v2
f’(x) = [(x+1) 2x – x2]/(x+1)2
= (2x2 + 2x – x2)/(x+1)2
= (x2+2x)/(x+1)2
= x(x+2)/(x+1)2
Matrix
We can define a matrix as an array of data in rows and columns. A matrix of order m×n shows that the given matrix has m rows and n columns. A unit matrix is a matrix in which all diagonal elements are 1, and other elements will be zero. Multiplication of two matrices is possible only if the number of columns of the first matrix is equal to the number of rows of the second matrix.
Matrix is an easy topic if learned properly. Students preparing for the JEE Main are recommended to learn the matrix thoroughly so that they can easily crack the questions from this topic and improve their ranks for the exam.