Question

find the derivative of $f(x)=\cot(2x^4 + 10x).$
$\bigcirc$ $f(x)=-\csc^2(8x^3 + 10)$
$\bigcirc$ $f(x)=(8x^3 + 10)\csc^2(2x^4 + 10x)$
$\bigcirc$ $f(x)=\csc^2(8x^3 + 10)$
$\bigcirc$ $f(x)=-(8x^3 + 10)\csc^2(2x^4 + 10x)$

Explanation:

Step1: Identify inner/outer functions

Let $u = 2x^4 + 10x$, so $f(x) = \cot(u)$.

Step2: Derive outer function

Derivative of $\cot(u)$ is $-\csc^2(u)$.

Step3: Derive inner function

Derivative of $u$: $\frac{du}{dx} = 8x^3 + 10$.

Step4: Apply chain rule

Multiply derivatives: $f'(x) = -\csc^2(u) \cdot \frac{du}{dx}$.
Substitute back $u = 2x^4 + 10x$:
$f'(x) = -(8x^3 + 10)\csc^2(2x^4 + 10x)$

Answer:

$f'(x) = -(8x^3 + 10)\csc^2(2x^4 + 10x)$ (the fourth option)