Chain Rule
One of the core principles in Calculus is the Chain Rule.
Refer to Khan academy article: Chain rule▶ Proceed to Integral rule of composite functions: U-substitution
It tells us how to differentiate Composite functions.
It must be composite functions, and it has to have inner & outer functions, which you could write in form of f(g(x)).
Common mistakes
- Not recognizing whether a function is composite or not
- Wrong identification of the inner and outer function
- Forgetting to multiply by the derivative of the inner function
- Computing
f(g(x))wrongly:
How to identify Composite functions
Seems a basic algebra101, but actually a quite tricky one to identify.
Refer to Khan lecture: Identifying composite functions
The core principle to identify it, is trying to re-write the function into a nested one: f(g(x)). If you could do this, it's composite, if not, then it's not one.
Examples
It’s a composite function, which the inner is cos(x) and outer is x².
It’s a composite function, which the inner is 2x³-4x and outer is sin(x).
It’s a composite function, which the inner is cos(x) and outer is √(x).
Two forms of Chain Rule
The general form of Chain Rule is like this:
But the Chain Rule has another more commonly used form:
Their results are exactly the same.
It’s just some people find the first form makes sense, some more people find the second one does.
Example
Solve:
Refer to Symbolab worked example.%5Cright)%5Cright))
Chain rule for exponential function
Formula:
Because:
Example
Solve:
- Apply the
Log power ruleto simplify the exponential function:
- Differentiate both sides:
