The problem is to differentiate y = (3sqrt(x) − 2x)3.

Question

ScanSolve AI Tutor · Accepted Answer

Okay, Ni_Fe_Mi✨. Let's differentiate this one.

The problem is to differentiate $y = (3\sqrt{x} - 2x)^3$ .

Rewrite the function using fractional exponents for the inner term.
$y = (3x^{1/2} - 2x)^3$
Apply the chain rule.
Let $u = 3x^{1/2} - 2x$ . The derivative of $y = u^3$ is $\frac{dy}{dx} = 3u^2 \cdot \frac{du}{dx}$ .
$\frac{dy}{dx} = 3(3x^{1/2} - 2x)^{3-1} \cdot \frac{d}{dx}(3x^{1/2} - 2x)$
Differentiate the inner function $\frac{d}{dx}(3x^{1/2} - 2x)$ .
$\frac{d}{dx}(3x^{1/2} - 2x) = 3 \cdot \frac{1}{2}x^{\frac{1}{2}-1} - 2 \cdot 1$ $= \frac{3}{2}x^{-1/2} - 2$
Substitute the derivative of the inner function back into the expression from Step 2 and simplify.
$\frac{dy}{dx} = 3(3x^{1/2} - 2x)^2 \left(\frac{3}{2}x^{-1/2} - 2\right)$
Rewrite the terms with positive and radical exponents for clarity.
$\frac{dy}{dx} = 3(3\sqrt{x} - 2x)^2 \left(\frac{3}{2\sqrt{x}} - 2\right)$
The derivative is $\boxed{\mathbf{3(3\sqrt{x} - 2x)^2 \left(\frac{3}{2\sqrt{x}} - 2\right)}}$ .

Step-by-step solution