Question

differentiate
⑤ $\frac{1}{sqrt{x^{3}+1}}$
⑥ $x^{2}5^{-3x}$

Explanation:

Step1: Rewrite the function

Rewrite \(\frac{1}{\sqrt{x^{3}+1}}\) as \((x^{3}+1)^{-\frac{1}{2}}\).

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 \(u = x^{3}+1\), so \(y = u^{-\frac{1}{2}}\). First, find \(\frac{dy}{du}\):

$$ \frac{dy}{du}=-\frac{1}{2}u^{-\frac{3}{2}} $$

Then find \(\frac{du}{dx}\):

$$ \frac{du}{dx}=3x^{2} $$

By the chain rule \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\).
Substitute \(u = x^{3}+1\) back in:

$$ \frac{dy}{dx}=-\frac{1}{2}(x^{3}+1)^{-\frac{3}{2}}\cdot3x^{2}=-\frac{3x^{2}}{2(x^{3}+1)^{\frac{3}{2}}} $$

Answer:

The derivative of \(\frac{1}{\sqrt{x^{3}+1}}\) is \(-\frac{3x^{2}}{2(x^{3}+1)^{\frac{3}{2}}}\)