Chain rule and implicit forms
When functions are stacked inside each other, differentiate the outside, leave the inside alone, then multiply by the inside's derivative.
The chain rule
If y is a function of u and u is a function of x, the derivative of the composition is the outside derivative evaluated at the inside, times the inside derivative. In words: differentiate outer, keep inner, multiply by derivative of inner. This handles composites like sin of 3x^2 in one step.
(f(g(x)))' = f'(g(x)) \cdot g'(x)Leibniz form
In Leibniz notation the chain rule reads dy/dx equals dy/du times du/dx, and the du terms cancel as if they were fractions. This form makes nested rates multiply, which is why it is the workhorse of related rates problems. Treat any intermediate variable as the u.
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}Implicit differentiation
For curves defined by an equation in x and y, differentiate both sides with respect to x, applying the chain rule to every y term so each contributes a dy/dx factor. Then solve algebraically for dy/dx. For the circle x^2 + y^2 = 25, differentiating gives 2x + 2y (dy/dx) = 0, so dy/dx = -x/y.
x^2 + y^2 = 25 \implies \frac{dy}{dx} = -\frac{x}{y}Flashcard
Chain rule for composite functions
Check your understanding
What is the derivative of g(x) = \sin(3x^2)?