let f(x)=3lnsin(x). then, f(x)=| submit answer next item
let f(x)=3lnsin(x). then, f(x)=| submit answer next item
Answer
Explanation:
Step1: Find first - derivative
Use chain - rule. If $y = 3\ln(u)$ and $u=\sin(x)$, then $\frac{dy}{du}=\frac{3}{u}$ and $\frac{du}{dx}=\cos(x)$. By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. So $f'(x)=\frac{3\cos(x)}{\sin(x)} = 3\cot(x)$.
Step2: Find second - derivative
Differentiate $f'(x)=3\cot(x)$ with respect to $x$. Since the derivative of $\cot(x)=-\csc^{2}(x)$, then $f''(x)=3\times(-\csc^{2}(x))=- 3\csc^{2}(x)$.
Answer:
$-3\csc^{2}(x)$