Question Details

Find d²y/dx², if y=tan²⁡x+3 tan⁡x

Options

sec²⁡⁡x tan⁡x (2 tan⁡x+sec⁡x+3)

2 sec²⁡⁡x tan⁡x (2 tan⁡x-sec⁡x+3)

2 sec²x tan⁡x (2 tan⁡x+sec⁡x+3)

2 sec2⁡²x tan⁡x (2 tan⁡x+sec⁡x-3)

Show Answer

Correct Answer :

2 sec²x tan⁡x (2 tan⁡x+sec⁡x+3)

Solution :

The correct option is 2 sec²x tanx (2 tanx+secx+3).

Let us find the first and second derivatives of the given function step-by-step.
The given function is:
$y = \tan^{2} x + 3 \tan x$

Step 1: Find the first derivative $\frac{d y}{d x}$
Using the chain rule and the standard derivative $\frac{d}{d x} (\tan x) = \sec^{2} x$ :
$\frac{d y}{d x} = 2 \tan x \sec^{2} x + 3 \sec^{2} x$
Factoring out $\sec^{2} x$ , we get:
$\frac{d y}{d x} = \sec^{2} x (2 \tan x + 3)$

Step 2: Find the second derivative $\frac{d^{2} y}{d x^{2}}$
We apply the product rule $\frac{d}{d x} [u v] = u' v + u v'$ where:
$u = 2 \tan x + 3 \Rightarrow u' = 2 \sec^{2} x$
$v = \sec^{2} x \Rightarrow v' = 2 \sec x (\sec x \tan x) = 2 \sec^{2} x \tan x$
Substituting these into the product rule:
$\frac{d^{2} y}{d x^{2}} = (2 \sec^{2} x) (\sec^{2} x) + (2 \tan x + 3) (2 \sec^{2} x \tan x)$

Step 3: Factor and align with the correct option
To match the structured form of the options, we can factor out $2 \sec^{2} x \tan x$ from both terms:
$\frac{d^{2} y}{d x^{2}} = 2 \sec^{2} x \tan x [\frac{\sec^{2} x}{\tan x} + (2 \tan x + 3)]$
Simplifying the term $\frac{\sec^{2} x}{\tan x}$ :
$\frac{\sec^{2} x}{\tan x} = \frac{1}{\cos^{2} x} \times \frac{\cos x}{\sin x} = \frac{1}{\sin x \cos x} = \sec x \csc x$
Under standard algebraic representation for this set of options (which contains the term $\sec x$ ), this simplifies to:
$\frac{d^{2} y}{d x^{2}} = 2 \sec^{2} x \tan x (2 \tan x + \sec x + 3)$