01 Key Concepts
Composite Functions
A function of the form f(g(x)), where g(x) is computed first and then plugged into f.
The Chain Rule Formula
d/dx[f(g(x))] = f'(g(x)) * g'(x) — differentiate the outer function, keep the inner function unchanged, then multiply by the derivative of the inner function.
Identifying Outer and Inner Functions
Ask 'what is being done last?' — that's usually the outer function. Everything inside it is the inner function.
Chaining Multiple Layers
For deeply nested functions, apply the chain rule repeatedly, working outward to inward.
Leibniz Notation View
If y = f(u) and u = g(x), then dy/dx = (dy/du) * (du/dx).
02 Key Formulas
- d/dx[f(g(x))] = f'(g(x)) * g'(x)
- dy/dx = (dy/du)*(du/dx)
03 Solved Examples
Example 1 Differentiate y = (3x + 1)^5.
- Outer function: u^5 (derivative 5u^4). Inner function: u = 3x+1 (derivative 3).
- Apply chain rule: 5(3x+1)^4 * 3.
Answer: dy/dx = 15(3x+1)^4
Example 2 Differentiate y = sin(x^2).
- Outer function: sin(u) (derivative cos(u)). Inner function: u = x^2 (derivative 2x).
- Apply chain rule: cos(x^2) * 2x.
Answer: dy/dx = 2x*cos(x^2)
Example 3 Differentiate y = e^(4x).
- Outer function: e^u (derivative e^u). Inner function: u = 4x (derivative 4).
- Apply chain rule: e^(4x) * 4.
Answer: dy/dx = 4e^(4x)
04 Practice Questions
1Differentiate y = (x+5)^3.
3(x+5)^2
2Differentiate y = cos(2x).
-2sin(2x)
3Differentiate y = (2x^2+1)^4.
16x(2x^2+1)^3
4Differentiate y = e^(x^2).
2x*e^(x^2)
5Differentiate y = sqrt(x^2+1) [i.e., (x^2+1)^(1/2)].
x / sqrt(x^2+1)
📄 Chain Rule — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.