find the derivative of the following function. y = -3 / (5x^5 + 8)^4 dy/dx = □

Explanation:

Step1: Rewrite the function

Rewrite $y = \frac{-3}{(5x^{5}+8)^{4}}$ as $y=-3(5x^{5}+8)^{-4}$.

Step2: Apply the chain - rule

The chain - rule states that if $y = f(g(x))$, then $y^\prime=f^\prime(g(x))\cdot g^\prime(x)$. Let $u = 5x^{5}+8$, so $y=-3u^{-4}$. First, find $\frac{dy}{du}$ and $\frac{du}{dx}$. $\frac{dy}{du}=-3\times(-4)u^{-5}=12u^{-5}$ and $\frac{du}{dx}=25x^{4}$.

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

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = 5x^{5}+8$ back in: $\frac{dy}{dx}=12(5x^{5}+8)^{-5}\times25x^{4}=\frac{300x^{4}}{(5x^{5}+8)^{5}}$.

Answer:

$\frac{300x^{4}}{(5x^{5}+8)^{5}}$