Question

find the derivative of x^3 - 3x^2

Explanation:

Step1: Apply power - rule for differentiation

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$. For the function $y=x^3-3x^2$, we differentiate each term separately.
For the first term $x^3$, using the power - rule with $n = 3$, we have $\frac{d}{dx}(x^3)=3x^{3 - 1}=3x^2$.
For the second term $-3x^2$, using the power - rule with $n = 2$ and the constant multiple rule (if $y = cf(x)$ where $c$ is a constant, then $y^\prime=cf^\prime(x)$), we have $\frac{d}{dx}(-3x^2)=-3\times\frac{d}{dx}(x^2)=-3\times2x^{2 - 1}=-6x$.

Step2: Combine the derivatives of the terms

The derivative of $y=x^3-3x^2$ is the sum of the derivatives of its terms. So $y^\prime=\frac{d}{dx}(x^3-3x^2)=\frac{d}{dx}(x^3)-\frac{d}{dx}(3x^2)=3x^2-6x$.

Answer:

$3x^2 - 6x$