Question

question find the derivative of h(x)=2cos^12(x). provide your answer below: h(x)=□

Explanation:

Step1: Recall chain - rule

The chain - rule states that if $y = f(g(x))$, then $y^\prime=f^\prime(g(x))\cdot g^\prime(x)$. Let $u = \cos(x)$, so $h(x)=2u^{12}$.

Step2: Differentiate outer function

Differentiate $y = 2u^{12}$ with respect to $u$. Using the power - rule $\frac{d}{du}(au^n)=nau^{n - 1}$, we have $\frac{d}{du}(2u^{12})=2\times12u^{11}=24u^{11}$.

Step3: Differentiate inner function

Differentiate $u=\cos(x)$ with respect to $x$. We know that $\frac{d}{dx}(\cos(x))=-\sin(x)$.

Step4: Apply chain - rule

By the chain - rule, $h^\prime(x)=\frac{d}{du}(2u^{12})\cdot\frac{du}{dx}$. Substitute $u = \cos(x)$ and the derivatives we found above: $h^\prime(x)=24\cos^{11}(x)\cdot(-\sin(x))=- 24\sin(x)\cos^{11}(x)$.

Answer:

$-24\sin(x)\cos^{11}(x)$