IB Maths AI 5.4 Notes

Core derivative formulas

This section extends differentiation to a wider range of functions. The main new derivatives are for $x^n$ where $n\in\mathbb{Q}$, $\sin x$, $\cos x$, $e^x$ and $\ln x$, together with the chain rule, product rule and quotient rule. First, let's remember the power rule:

$\frac{d}{dx}(x^n)=nx^{n-1}$

From this, there are a few common derivatives you need to memorise:

• If $f(x)=\sin x$, then $f'(x)=\cos x$.
• If $f(x)=\cos x$, then $f'(x)=-\sin x$.
• If $f(x)=e^x$, then $f'(x)=e^x$.
• If $f(x)=\ln x$, then $f'(x)=\frac{1}{x}$ for $x\gt 0$.

These can be combined using sums and constant multiples for more complicated derivatives:

Differentiate $f(x)=3x^4-2\sin x+5e^x$.

For this, we differentiate term by term.

The result is $f'(x)=12x^3-2\cos x+5e^x$.

Note that the power rule also works for fractional powers. For example:

• If $f(x)=x^{\frac{1}{2}}$, then $f'(x)=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}$.
• If $f(x)=x^{-\frac{3}{2}}$, then $f'(x)=-\frac{3}{2}x^{-\frac{5}{2}}$.