The inverse trigonometric derivatives, reading the table backwards to get integrals, and when to reach for partial fractions.
This sub-topic is largely a table to know. So check it rather than trusting it: pick a function and move x, and the stated derivative is compared with the gradient measured from a tiny chord.
y = tan x → sec²x
| f(x) | tan x | sec x | cosec x | cot x |
|---|---|---|---|---|
| f′(x) | sec²x | sec x tan x | −cosec x cot x | −cosec²x |
| f(x) | ax | logax | arcsin x | arccos x | arctan x |
|---|---|---|---|---|---|
| f′(x) | ax ln a | 1x ln a | 1√(1−x²) | −1√(1−x²) | 11+x² |
Three patterns worth noticing. Every "co" function picks up a minus sign. arcsin and arccos differ only by that sign, which is why their graphs are mirror images. And arctan has no root in it, which makes 1⁄1+x² instantly recognisable as something that integrates to an arctan.
Every line is also an integral. The two that come up constantly:
The signal: a fraction whose bottom factorises, and whose top is not the derivative of the bottom. Split it, and each piece becomes a logarithm.
Factorise before deciding. If the bottom factorises, use partial fractions. If it does not, complete the square and expect an arctan. Deciding which one you are looking at is most of the work, and it takes ten seconds.
1. What is ddx(arctan x) at x = 1?
2. For ∫ 1x² − 4 dx, what should you do first?
3. ∫ sec²(2x + 5) dx is:
Recognising which of the three routes a fraction needs: reverse chain rule, partial fractions, or completing the square. That decision is usually the first mark and it determines everything after it.
Setting up the partial fractions correctly, and showing the working that finds the numerators.
The dividing constant on any composite with a linear function, and the modulus inside every logarithm.
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