(3x + 5) (1 + tan x)
Question:

(3x + 5) (1 + tan x)

Solution:

Given $(3 x+5)(1+\tan x)$

Let $y=(3 x+5)(1+\tan x)$

Applying product rule of differentiation that is

$\Rightarrow \frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{t} \cdot \mathrm{y})=\mathrm{y} \cdot \frac{\mathrm{d} \mathrm{t}}{\mathrm{dx}}+\mathrm{t} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}$

$\Rightarrow y=(3 x+5)(1+\tan x)$

$\Rightarrow \frac{d y}{d x}=(1+\tan x) \frac{d}{d x}(3 x+5)+(3 x+5) \frac{d}{d x}(1+\tan x)$

$\Rightarrow \frac{d y}{d x}=3(1+\tan x)+(3 x+5) \sec ^{2} x$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=3 \mathrm{x} \sec ^{2} \mathrm{x}+5 \sec ^{2} \mathrm{x}+3+3 \tan \mathrm{x}$ (by using product rule)

Hence, the required answer is $3 x \sec ^{2} x+5 \sec ^{2} x+3 \tan x+3$