d/dx (-2 + x^2y) equals 2xy + x^2y -2 + 2xy + x^2y -2 + 2xy + x^2y 2xy + x^2y

Explanation:

Step1: Apply sum - difference rule

$\frac{d}{dx}(-2 + x^{2}y)=\frac{d}{dx}(-2)+\frac{d}{dx}(x^{2}y)$

Step2: Differentiate constant

Since the derivative of a constant $c$ with respect to $x$ is 0, $\frac{d}{dx}(-2)=0$.

Step3: Apply product rule

The product rule states that if $u = x^{2}$ and $v = y$, then $\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$. Here, $\frac{du}{dx}=\frac{d}{dx}(x^{2}) = 2x$ and $\frac{dv}{dx}=y'$. So $\frac{d}{dx}(x^{2}y)=x^{2}y'+y\cdot2x$.

Step4: Combine results

$\frac{d}{dx}(-2)+\frac{d}{dx}(x^{2}y)=0 + 2xy+x^{2}y'=2xy + x^{2}y'$

Answer:

$2xy + x^{2}y'$