13. 0 / 1.33 points 1/2 submissions used find the derivative of the function. g(z)=(6 + cos²(z))³ g(z)=-3…

13. 0 / 1.33 points 1/2 submissions used find the derivative of the function. g(z)=(6 + cos²(z))³ g(z)=-3 sin(z)(6 + cos²(z))²

13. 0 / 1.33 points 1/2 submissions used find the derivative of the function. g(z)=(6 + cos²(z))³ g(z)=-3 sin(z)(6 + cos²(z))²

Answer

Explanation:

Step1: Apply chain - rule

Let $u = 6+\cos^{2}(z)$. Then $G(z)=u^{3}$. The chain - rule states that $\frac{dG}{dz}=\frac{dG}{du}\cdot\frac{du}{dz}$. First, find $\frac{dG}{du}$. Since $G(u) = u^{3}$, by the power rule $\frac{dG}{du}=3u^{2}=3(6 + \cos^{2}(z))^{2}$.

Step2: Find $\frac{du}{dz}$

Let $v=\cos(z)$. Then $u = 6 + v^{2}$. First, $\frac{du}{dv}=2v = 2\cos(z)$ and $\frac{dv}{dz}=-\sin(z)$. By the chain - rule, $\frac{du}{dz}=\frac{du}{dv}\cdot\frac{dv}{dz}=2\cos(z)\cdot(-\sin(z))=- 2\sin(z)\cos(z)$.

Step3: Calculate $\frac{dG}{dz}$

By the chain - rule $\frac{dG}{dz}=\frac{dG}{du}\cdot\frac{du}{dz}=3(6 + \cos^{2}(z))^{2}\cdot(-2\sin(z)\cos(z))=-6\sin(z)\cos(z)(6 + \cos^{2}(z))^{2}$.

Answer:

$-6\sin(z)\cos(z)(6 + \cos^{2}(z))^{2}$