## Task Condition

Find the derivative.

$$
y=\left(x^{4}+5\right)^{\cot x}
$$
## Solution

$y=\left(x^{4}+5\right)^{\cot x}$

$\ln y=\ln \left(x^{4}+5\right)^{\cot x}=\cot x \cdot \ln \left(x^{4}+5\right)$
$\frac{y'}{y}=\frac{-1}{\sin ^{2} x} \cdot \ln \left(x^{4}+5\right)+\cot x \cdot \frac{1}{x^{4}+5} \cdot 4 x^{3}=$

$=\frac{4 x^{3} \cdot \cot x}{x^{4}+5}-\frac{\ln \left(x^{4}+5\right)}{\sin ^{2} x}$

$y'=y \cdot\left(\frac{4 x^{3} \cdot \cot x}{x^{4}+5}-\frac{\ln \left(x^{4}+5\right)}{\sin ^{2} x}\right)=$

$=\left(x^{4}+5\right)^{\cot x} \cdot\left(\frac{4 x^{3} \cdot \cot x}{x^{4}+5}-\frac{\ln \left(x^{4}+5\right)}{\sin ^{2} x}\right)$

Kuznetsov Differentiation 12-16