calculate the derivative of the following function. y = 8(6x^3 + 7)^(-6) dy/dx = □

Explanation:

Step1: Apply chain - rule

Let $u = 6x^{3}+7$, then $y = 8u^{-6}$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$. $\frac{dy}{du}=\frac{d}{du}(8u^{-6})=8\times(-6)u^{-7}=-48u^{-7}$

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

$\frac{du}{dx}=\frac{d}{dx}(6x^{3}+7)=6\times3x^{2}+0 = 18x^{2}$

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

Substitute $u = 6x^{3}+7$ into $\frac{dy}{du}$ and multiply by $\frac{du}{dx}$: $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=-48(6x^{3}+7)^{-7}\times18x^{2}=-864x^{2}(6x^{3}+7)^{-7}$

Answer:

$-864x^{2}(6x^{3}+7)^{-7}$