Question

attempt 1: 10 attempts remaining. compute the derivative of the function $y = sqrt{x^{3}-90x}$ using the chain rule. $\frac{dy}{dx}=$

Explanation:

Step1: Rewrite the function

Rewrite $y = \sqrt{x^{3}-90x}=(x^{3}-90x)^{\frac{1}{2}}$.

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 $u = x^{3}-90x$, so $y = u^{\frac{1}{2}}$. First, find the derivative of $y$ with respect to $u$: $\frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}}$. Then find the derivative of $u$ with respect to $x$: $\frac{du}{dx}=3x^{2}-90$.

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

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}$ and $\frac{du}{dx}$: $\frac{dy}{dx}=\frac{1}{2}(x^{3}-90x)^{-\frac{1}{2}}\cdot(3x^{2}-90)$.

Step4: Simplify the result

$\frac{dy}{dx}=\frac{3x^{2}-90}{2\sqrt{x^{3}-90x}}$.

Answer:

$\frac{3x^{2}-90}{2\sqrt{x^{3}-90x}}$