Q.Differentiate $y = x^3+5x^2-7x+2$ with respect to $x$.

This is a sum of simple power terms, so the sum/difference rule lets us differentiate it term by term using the power rule.

Step 1 — Apply the sum/difference rule. Since $y$ is a sum and difference of four terms, differentiate each term separately.

Step 2 — Differentiate each term using the power rule ($\frac{d}{dx}x^n=nx^{n-1}$) and the constant rule.

• $\dfrac{d}{dx}(x^3) = 3x^2$
• $\dfrac{d}{dx}(5x^2) = 5\cdot2x = 10x$
• $\dfrac{d}{dx}(-7x) = -7$ (since $\frac{d}{dx}x=1$)
• $\dfrac{d}{dx}(2) = 0$ (derivative of a constant)

Step 3 — Combine. $\dfrac{dy}{dx} = 3x^2+10x-7+0 = 3x^2+10x-7$. …

Forgetting to bring down and multiply the existing coefficient correctly (e.g. differentiating $5x^2$ as $5x$ instead of $10x$) is the most common slip. Another is forgetting that a lone constant …