1. Why the Chain Rule Is Needed
The chain rule helps differentiate composite functions — functions where one function is inside another. Many real expressions are not simple sums or products but contain nested parts. The chain rule breaks such expressions into an ‘outer function’ and an ‘inner function’.
2. Idea Behind the Chain Rule
The derivative of a composite function changes depending on both the outer function and how fast the inner function is changing. The chain rule captures this combined effect.
In notes-style thinking: differentiate the outside part first, keep the inside as it is, then multiply by the derivative of the inside.
3. Chain Rule Formula
\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
This formula is the heart of the chain rule and works for any composition.
4. Identifying Outer and Inner Functions
To use the chain rule comfortably, it's helpful to separate the expression into layers.
4.1. Examples of Identifying Layers
- For \( (x^2 + 1)^5 \): inner = x^2 + 1, outer = t^5
- For \( \sin(3x) \): inner = 3x, outer = sin(t)
- For \( e^{x^3} \): inner = x^3, outer = e^t
5. Examples Using the Chain Rule
Here are a few clean examples applying the rule directly.
5.1. Example 1 — Power of a Function
Differentiate:
f(x) = (x^2 + 1)^5
Outer function = t^5 → derivative = 5t^4
Inner function = x^2 + 1 → derivative = 2x
f'(x) = 5(x^2 + 1)^4 \cdot 2x = 10x(x^2 + 1)^4
5.2. Example 2 — Trigonometric Case
Differentiate:
f(x) = \sin(3x)
Outer derivative = cos
Inner derivative = 3
f'(x) = \cos(3x) \cdot 3 = 3 \cos(3x)
5.3. Example 3 — Exponential Case
Differentiate:
f(x) = e^{x^3}
Outer derivative = e^t
Inner derivative = 3x^2
f'(x) = e^{x^3} \cdot 3x^2
5.4. Example 4 — Logarithmic Case
Differentiate:
f(x) = \ln(5x + 2)
Outer derivative = 1/t
Inner derivative = 5
f'(x) = \frac{1}{5x + 2} \cdot 5 = \frac{5}{5x + 2}
6. Recognising Chain Rule Situations
The chain rule appears everywhere — exponentials with powers, logs with functions inside, trigonometric functions with modified angles, or any expression that has an inner structure. Once you start spotting the inner layer, the rule becomes natural to apply.