The chain rule
- The chain rule differentiates a function that is inside another function, like . You do not have to expand seven brackets.
- Use it for tangents to curves with brackets, for stationary points, and for every rate-of-change question later on.
Outside, then inside. Differentiate the outer function and leave the inside as it is. Then multiply by the derivative of the inside: . That second part is the bit most people forget.
The shortcut you will use most: . The inside derivative is just the number . So brings down a factor of .
Name the inside, then multiply
- For , let . Now the whole thing is , a plain power.
The chain rule joins the two derivatives by multiplying them:
Worked example
Differentiate
- Outside (the power): bring the down, drop the power by one — .
- Inside: .
- Multiply the two: .
Roots and reciprocals too
- The same steps work for roots and fractions once you write them as powers.
- A function like becomes . The inside is (its derivative is ).
Worked example
Differentiate
- Write as a power so the rule applies: .
- Outside: . Inside: .
- Multiply: .
Why you multiply the two derivatives, not add them
Think of gears. Say changes twice as fast as (). Say changes three times as fast as (). Then changes times as fast as . The two rates multiply. That is why you multiply the two factors. It is the same idea as connected rates of change in §7.6.
Where the marks go
Common mistake
Common mistake
Now you try
The equation of a curve is . Find . (9709/12 May/June 2020 Q10)
Your turn— tap to reveal the worked answer (9709/12 May/June 2020 Q10(a))
The first term is easy: .
For : outside , inside , so .