Differentiation Rules: Overview, Questions, Preparation

Differentiation is the process of finding the derivatives of a function. In Mathematics, it is defined as the derivative of a function for an independent variable.

Some of the basic differentiation formulas are stated below:

Sum and Difference Rule:

The derivative of a function is the sum of the differences of the single function, which is 

F(x) = u(x) + v(x)

Product Rule: 
In this rule, one function f(x) is the product of two functions u(x) and v(x), and the derivative of a function is

F(x) = u(x) * v(x)

Then f’(x) = u’(x) * v’(x) + u’(x) * v’(x)

Quotient Rule: 

In this rule, the function f(x) is in the form of two functions [u(x)]/[v(x)], the derivative of function is

F(x) = u’(x) * v’(x) - u’(x) * v’(x)/(v(x))²

Chain Rule:

In chain rule, the derivative of the function y = f(x) = g(u) and if u = h(x), then

Power Rule:

If x is a variable and y is another variable in the power rule, then the rate of change of x with respect to y is given by dy/dx. 

Chapter Continuity and Differentiability includes the topic of differential rules. The students learn about differentiating certain functions like polynomial and trigonometric functions and various differentials rules.

1. Prove that the function f given by

 f(x) = | x – 1|, x ∈ R
 is not differentiable at x = 1.

Solution:

Given, f(x) = | x – 1|
Then f(1) = |1 - 1| = 0
Right hand limit: f’(1) = f1+h-f1/h    
= /h 

Left hand limit ≠ Right hand limit.

2. Find dy/ dx in the 2x + 3y = sin x

Solution:

Given, 2x + 3y = sin x

3. Find dy/ dx in the  y^x = x^y 

Solution:

Given, y^x = x^y 

logx^y = log y^x

y log x = x log y

Q: Define Differentiation?

Q: What is a linear and non-linear function?

Q: What is the meaning of continuity of a function?

Q: What is the constant rule of differentiation?

Q: What is the product rule of differentiation?