Question

Differentiate the function given below w.r.t. $x:$

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

Answer 1

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

$\frac{dy}{dx}=\frac{d}{dx}\left[\right(3x+5\left)\right(1+\tan x\left)\right]$

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

$[\because \frac{d(u\cdot v)}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}]$

$\Rightarrow \frac{dy}{dx}=(3x+5)[\frac{d\left(1\right)}{dx}+\frac{d(\tan x}{dx}]+$

$(1+\tan x)[\frac{d\left(3x\right)}{dx}+\frac{d\left(5\right)}{dx}]$

$[\because \frac{d(f+g)}{dx}=\frac{df}{dx}+\frac{dg}{dx}]$

$\Rightarrow \frac{dy}{dx}=(3x+5)(0+{\sec }^{2}x)+(1+\tan x)(3+0)$

$[\frac{d(\tan x}{dx}={\sec }^{2}x,\frac{d\left(c\right)}{dx}=0]$

$\frac{dy}{dx}=(3x+5){\sec }^{2}x+3(1+\tan x)$