Question

attempt 1: 10 attempts remaining. compute the derivative of the function ( y = sqrt{x^3 - 30x} ) using the chain rule. ( \frac{dy}{dx} = ) input box pencil icon submit answer next item

Explanation:

Step1: Rewrite the function

Rewrite \( y = \sqrt{x^3 - 30x} \) as \( y=(x^3 - 30x)^{\frac{1}{2}} \).

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 \( u=x^3 - 30x \) and \( n=\frac{1}{2} \).
First, find \( \frac{du}{dx} \): \( \frac{du}{dx}=\frac{d}{dx}(x^3 - 30x)=3x^2-30 \).
Then, find \( \frac{dy}{du} \): \( \frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}}=\frac{1}{2}(x^3 - 30x)^{-\frac{1}{2}} \) (substituting back \( u = x^3 - 30x \)).
Now, multiply \( \frac{dy}{du} \) and \( \frac{du}{dx} \) to get \( \frac{dy}{dx} \):
\( \frac{dy}{dx}=\frac{1}{2}(x^3 - 30x)^{-\frac{1}{2}}\cdot(3x^2 - 30) \)
Simplify the expression:
\( \frac{dy}{dx}=\frac{3x^2 - 30}{2\sqrt{x^3 - 30x}}=\frac{3(x^2 - 10)}{2\sqrt{x^3 - 30x}} \) (factoring out 3 from the numerator)

Answer:

\( \frac{3x^2 - 30}{2\sqrt{x^3 - 30x}} \) (or the factored form \( \frac{3(x^2 - 10)}{2\sqrt{x^3 - 30x}} \))