Every rule for finding a derivative
Which differentiation rule actually applies to the shape in front of you.
Read the expression's outermost structure first. A single power of x (or a constant multiple/sum of them) is the power rule. Two factors multiplied together is the product rule; one function divided by another is the quotient rule. A function plugged inside another function is the chain rule — and it composes with the other three, so a product or quotient whose pieces are themselves composite functions needs the chain rule applied inside the product/quotient rule, not instead of it. An equation that isn't already solved for y calls for implicit differentiation. Everything else in this course's differentiation toolkit is one of these five ideas combined, never a sixth technique.
From first principles
Term
derivative
Every rule below is a shortcut derived from this same difference-quotient limit.
The basic shapes
For any real exponent n, \frac{d}{dx}\left[x^n\right] = nx^{n-1}. A constant multiple and a sum of terms each differentiate independently: \dfrac{d}{dx}[cf(x)] = c f'(x) and \dfrac{d}{dx}[f(x)+g(x)] = f'(x)+g'(x).
Example
Example
Example
A function inside a function
If y = f(u) and u = g(x), then \frac{dy}{dx} = f'(g(x)) \cdot g'(x).
Example
When y isn't isolated
Example