Question

find the derivative of y = 2x²(3x - 4)

Explanation:

Step1: Expand the function

$$y = 2x^{2}(3x - 4)=6x^{3}-8x^{2}$$

Step2: Apply the power rule

The power rule is $\frac{d}{dx}(x^{n})=nx^{n - 1}$.
For $y = 6x^{3}-8x^{2}$, the derivative $\frac{dy}{dx}$:

• For the term $6x^{3}$: Using the power rule, $\frac{d}{dx}(6x^{3})=6\times3x^{3 - 1}=18x^{2}$
• For the term $-8x^{2}$: Using the power rule, $\frac{d}{dx}(-8x^{2})=-8\times2x^{2 - 1}=-16x$

Step3: Combine the derivatives of the terms

$$\frac{dy}{dx}=18x^{2}-16x$$

Answer:

$18x^{2}-16x$ (the fourth option)