Question

differentiate. y = 3x^3 - 10x^2 + 32x + 6

Explanation:

Step1: Apply power - rule to each term

The power - rule for differentiation is $\frac{d}{dx}(ax^n)=nax^{n - 1}$. For the term $3x^3$, $n = 3$ and $a = 3$, so $\frac{d}{dx}(3x^3)=3\times3x^{3 - 1}=9x^2$. For the term $-10x^2$, $n = 2$ and $a=-10$, so $\frac{d}{dx}(-10x^2)=2\times(-10)x^{2 - 1}=-20x$. For the term $32x$, $n = 1$ and $a = 32$, so $\frac{d}{dx}(32x)=1\times32x^{1 - 1}=32$. For the constant term $6$, $\frac{d}{dx}(6)=0$ since the derivative of a constant is $0$.

Step2: Combine the derivatives of each term

$\frac{dy}{dx}=\frac{d}{dx}(3x^3)-\frac{d}{dx}(10x^2)+\frac{d}{dx}(32x)+\frac{d}{dx}(6)=9x^2-20x + 32+0=9x^2-20x + 32$

Answer:

$\frac{dy}{dx}=9x^2-20x + 32$