If y= ^{-1}≤ft({12 x-64 x³}{1-48 x²}), then d y/d x=

Correct Answer is : 121+16x2{12}{1+16 x²}1+16x212

Correct Answer is : 121+16x2{12}{1+16 x²}1+16x212

Test Index

MHTCET 25 Apr 2025 Shift 2 Paper

Question : 116 of 150

Marks: +1, -0

y=\tan ^{-1}\left(\frac{12 x-64 x^{3}}{1-48 x^{2}}\right)

\frac{d y}{d x}=

Solution:

Concept:

Use the substitution

4x=\tan\theta

to simplify the expression into the standard form

\tan^{-1}(\tan 3\theta)

Explanation:

Let

4x=\tan\theta

, which gives

\theta=\tan^{-1}(4x)

Substitute

4x=\tan\theta

in the given expression:

y=\tan^{-1}\left(\frac{3\tan\theta-\tan^3\theta}{1-3\tan^2\theta}\right)

Using the identity

\tan 3\theta=\frac{3\tan\theta-\tan^3\theta}{1-3\tan^2\theta}

, we obtain:

y=\tan^{-1}(\tan 3\theta)=3\theta

Therefore,

y=3\tan^{-1}(4x)

Differentiate with respect to

x

\frac{dy}{dx}=3\cdot\frac{1}{1+(4x)^2}\cdot4=\frac{12}{1+16x^2}

Answer:

\frac{dy}{dx}=\frac{12}{1+16x^2}

Hence, the correct option is C.

Go to Question:

Prev Question

Next Question