Question

Differentiate w.r.t. $x$:

$e^{\sin x}+(\tan x{)}^x$

Answer 1

$y=e^{\sin x}+(\tan x{)}^x$

Let $z=(\tan x{)}^x$

Taking log on both sides

$\log z=x\log \tan x$

Differentiating w.r.t. x, we get,

$\frac{1}{z}\frac{dz}{dx}=\log \tan x+\frac{x{\sec }^{2}x}{\tan x}$

$\frac{dz}{dx}=\left(\tan x{)}^x\right[\log \tan x+\frac{x{\sec }^{2}x}{tanx}]$

Thus, $\frac{dy}{dx}=e^{\sin x}\cos x+\left(\tan x{)}^x\right[\log \tan x+\frac{x{\sec }^{2}x}{\tan x}]$