Question

The derivative of $(\cot x{)}^{\sin x}+(\tan x{)}^{\cos x}$ with respect to $x$ is

• A. $(\cot x{)}^{\sin x}[\sec x+\cos x\log (\cot x)]+(\tan x{)}^{\cos x}[\mathrm{cosec}x-\sin x\log (\tan x)]$
• B. $(\cot x{)}^{\sin x}[-\sec x+\cos x\log (\cot x)]+(\tan x{)}^{\cos x}[\cos ecx+\sin x\log (\tan x)]$
• C. $(\cot x{)}^{\sin x}[\sec x-\cos x\log (\cot x)]+(\tan x{)}^{\cos x}[\mathrm{cosec}x-\sin x\log (\tan x)]$
• D. $(\cot x{)}^{\sin x}[-\sec x+\cos x\log (\cot x)]+(\tan x{)}^{\cos x}[\cos ecx-\sin x\log (\tan x)]$

Answer 1

Answer: (D)

$\frac{d}{dx}\left[f(x{)}^{g(x)}\right]=f(x{)}^{g(x)}\left[g(x)\frac{f^{{}^{\prime }}(x)}{f(x)}+g^{{}^{\prime }}(x)\log (f(x))\right]$