let f(x)=3lnsin(x). then, f(x)=3csc²(x) submit answer next item
let f(x)=3lnsin(x). then, f(x)=3csc²(x) submit answer next item
Answer
Explanation:
Step1: Find first - derivative
Use the chain - rule. If $y = 3\ln(u)$ and $u=\sin(x)$, then $\frac{dy}{du}=\frac{3}{u}$ and $\frac{du}{dx}=\cos(x)$. So, $f^\prime(x)=\frac{3\cos(x)}{\sin(x)} = 3\cot(x)$.
Step2: Find second - derivative
Differentiate $f^\prime(x)=3\cot(x)$ with respect to $x$. Since the derivative of $\cot(x)=-\csc^{2}(x)$, then $f^{\prime\prime}(x)=3\times(- (-\csc^{2}(x)))=3\csc^{2}(x)$.
Answer:
$3\csc^{2}(x)$