Question

find the derivative of the function.
y = (7cos(x^6) - 1)^5
y =

Explanation:

Step1: Apply chain - rule

Let $u = 7\cos(x^{6})-1$, then $y = u^{5}$. By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$.
$\frac{dy}{du}=5u^{4}=5(7\cos(x^{6}) - 1)^{4}$

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

Let $v=x^{6}$, then $u = 7\cos(v)-1$. First find $\frac{du}{dv}$ and $\frac{dv}{dx}$. $\frac{du}{dv}=-7\sin(v)=-7\sin(x^{6})$ and $\frac{dv}{dx}=6x^{5}$. By the chain - rule $\frac{du}{dx}=\frac{du}{dv}\cdot\frac{dv}{dx}=-7\sin(x^{6})\cdot6x^{5}=-42x^{5}\sin(x^{6})$

Step3: Calculate $\frac{dy}{dx}$

$\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=5(7\cos(x^{6}) - 1)^{4}\cdot(-42x^{5}\sin(x^{6}))=-210x^{5}\sin(x^{6})(7\cos(x^{6}) - 1)^{4}$

Answer:

$-210x^{5}\sin(x^{6})(7\cos(x^{6}) - 1)^{4}$