Question

Differentiate log(sin(x^2+4)) with respect to x.

Answer 1

Given function,

$\longrightarrow f(x)=\log(\sin(x^2+4)$

Its derivative wrt $x$ will be,

$\longrightarrow f'(x)=\dfrac{d[\log(\sin(x^2+4)]}{dx}$

By chain rule,

$\longrightarrow f'(x)=\dfrac{d[\log(\sin(x^2+4))]}{d[\sin(x^2+4)]}\cdot\dfrac{d[\sin(x^2+4)]}{d[x^2+4]}\cdot\dfrac{d[x^2+4]}{dx}$

We have,

• $\dfrac{d}{dx}[\log x]=\dfrac{1}{x}$

• $\dfrac{d}{dx}[\sin x]=\cos x$

• $\dfrac{d}{dx}[x^2]=2x$

Then,

$\longrightarrow f'(x)=\dfrac{1}{\sin(x^2+4)}\cdot\cos(x^2+4)\cdot2x$

$\longrightarrow\underline{\underline{f'(x)=2x\cot(x^2+4)}}$