The Rules of Differentiation

Starting point: Years 9 and 10

The rules work on powers of x, so first rewrite everything as a power. This uses the index laws.

You also need two facts about straight lines:

• The gradient of $y=3x+2$ is 3, everywhere.
• The gradient of $y=7$ (a horizontal line) is 0.

These two facts are already derivatives: $\frac{d}{dx}(3x+2)=3$ and $\frac{d}{dx}(7)=0$.

Quick summary

• Rewrite roots and fractions as powers of $x$ before differentiating.
• Power rule: $\frac{d}{dx}{x}^n=n{x}^{n-1}$. Constants on their own vanish.
• Product: $u^{\prime }v+u{v}^{\prime }$. Quotient: $\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$. Chain: outside derivative times inside derivative.
• $e^x\to e^x$, $\ln x\to \frac{1}{x}$, $\sin x\to \cos x$, $\cos x\to -\sin x$, $\tan x\to {\sec }^{2}x$.

Quiz