Question

attempt 1: 10 attempts remaining. compute the derivative of the function ( y = sqrt{x^5 - 72x} ) using the chain rule. ( \frac{dy}{dx} = ) blank submit answer next item

Explanation:

Step1: Rewrite the function

Rewrite \( y = \sqrt{x^5 - 72x} \) as \( y=(x^5 - 72x)^{\frac{1}{2}} \).

Step2: Apply the Chain Rule

The Chain Rule states that if \( y = u^n \) and \( u = f(x) \), then \( \frac{dy}{dx}=n\cdot u^{n - 1}\cdot\frac{du}{dx} \). Let \( u=x^5 - 72x \) and \( n=\frac{1}{2} \).
First, find the derivative of \( u \) with respect to \( x \): \( \frac{du}{dx}=5x^4-72 \).
Then, find the derivative of \( y \) with respect to \( u \): \( \frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}} \).

Step3: Multiply the derivatives

Using the Chain Rule, \( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} \). Substitute the values of \( \frac{dy}{du} \) and \( \frac{du}{dx} \):
\[

$$\begin{align*} \frac{dy}{dx}&=\frac{1}{2}(x^5 - 72x)^{-\frac{1}{2}}\cdot(5x^4 - 72)\\ &=\frac{5x^4 - 72}{2\sqrt{x^5 - 72x}} \end{align*}$$

\]

Answer:

\(\frac{5x^{4}-72}{2\sqrt{x^{5}-72x}}\)