Find dy/dx: y = tan^-1 ( 3x - x³)/(1 - 3x²) , -1/√3 < x < 1/√3

Solution:

A derivative helps us to know the changing relationship between two variables. Consider the independent variable 'x' and the dependent variable 'y'.

The change in the value of the dependent variable with respect to the change in the value of the independent variable expression can be found using the derivative formula.

Given,

y = tan^-1 ( 3x - x³)/(1 - 3x²)

Let us find the derivative on both sides with respect to x.

We will proceed by using the inverse trigonometric concept.

Let us assume x = tan θ

⇒ θ = tan^-1 x. -----(1)

On substituting in the given function, we get

⇒ y = tan^-1 ( 3 tan θ - tan³ θ) / (1 - 3 tan² θ)

⇒ y = tan^-1 (tan 3θ) = 3θ

Since, tan 3θ = ( 3 tan θ - tan³ θ) / (1 - 3 tan² θ).

⇒ y = 3 θ = 3 tan^-1 x. [from 1]

On differentiating both sides wrt x, we get

dy/dx = 3 / (1 + x²)

Therefore,

the derivative of y = tan^-1 ( 3x - x³)/(1 - 3x²) is 3 / (1 + x²)

NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.3 Question 10