Chain Rule
Last updated
Last updated
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.
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:
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.
Two forms of Chain RuleTheir results are exactly the same. It's just some people find the first form makes sense, some more people find the second one does.
Chain rule for exponential functionApply the Log power rule to simplify the exponential function:
Differentiate both sides:
It must be composite functions, and it has to have inner & outer functions, which you could write in form of f(g(x)).
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).
The general form of Chain Rule is like this:
But the Chain Rule has another more commonly used form:
Solve: [Refer to Symbolab worked example.](https://www.symbolab.com/solver/step-by-step/\frac{d}{dx}\left(sqrt\left(3cos^{3}\left(x\right)\right)\right))
Formula:
Because:
Solve:
