Question

If \(y = 4\sqrt{x}\) then \(\frac{d^2y}{dx^2} =\)

Hint:

Remember: To find the second derivative, differentiate the first derivative once more using the power rule.

Answer 1

Step 1: Differentiate \(y = 4\sqrt{x}\).
First, rewrite \(y = 4x^{1/2}\). Now, differentiate with respect to \(x\): \[ \frac{dy}{dx} = \frac{d}{dx}(4x^{1/2}) = 4 \times \frac{1}{2} x^{-1/2} = \frac{2}{\sqrt{x}} \]
Step 2: Find the second derivative.
Differentiate \(\frac{dy}{dx} = \frac{2}{\sqrt{x}} = 2x^{-1/2}\) again: \[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left( 2x^{-1/2} \right) = 2 \times \left( -\frac{1}{2} \right) x^{-3/2} = -\frac{1}{x^{3/2}} \] Thus, the second derivative is: \[ \frac{d^2y}{dx^2} = -\frac{1}{x^{3/2}} \]