Question

What is the second derivative of tan 2x?
1. 8 tan² 2x sec² 2x
2. 8 tan 2x sec 2x
3. 8 tan 2x sec² 2x
4. None of these.

Answer 1

Correct Answer - Option 3 : 8 tan 2x sec² 2x

Concept:

Chain Rule of Derivatives:

• \(\rm \frac{d}{dx}f(g(x))=\frac{d}{d\ g(x)}f(g(x))\times \frac{d}{dx}g(x)\).
• \(\rm \frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}\).

 

Derivatives of Trigonometric Functions:

\(\rm \frac{d}{dx}\sin x=\cos x\ \ \ \ \ \ \ \ \ \ \ \ \ \frac{d}{dx}\cos x=-\sin x\\ \frac{d}{dx}\tan x=\sec^2x\ \ \ \ \ \ \ \ \ \ \ \frac{d}{dx}\cot x=-\csc^2 x\\ \frac{d}{dx}\sec x=\tan x\sec x\ \ \ \ \frac{d}{dx}\csc x=-\cot x\csc x\)

 

Calculation:

Using the chain rule of derivatives, we get:

\(\rm \frac{d\tan 2x}{dx} =\rm \frac{d\tan 2x}{d (2x)} \times \frac {d(2x)}{dx}\) = 2 sec² 2x

Again differentiation in respect to x, we get

\(\rm \frac{d^2\tan 2x}{dx^2} = \frac d {dx} \times \frac {d \tan 2x}{dx}\)

= \(\rm \frac {2d\sec^2 2x}{dx}\)

=2 (2 sec 2x)(tan 2x sec 2x)(2)

= 8 tan 2x sec² 2x