Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for AP calculus AB chain rule implicit differentiation? Then this AP Calculus AB lesson on Chain Rule and Implicit Differentiation explains each idea step by step, with worked examples.
Quick answer: The chain rule finds the derivative of a composite function by multiplying the derivative of the outer function, evaluated at the inner function, by the derivative of the inner function. Implicit differentiation extends this idea to equations where y is not isolated, and inverse formulas handle reflected graphs.
Standards: AP Calculus AB Unit 3 · Level: Grade 12 AP Calculus AB
How does the chain rule differentiate a composition?
If y = f(g(x)), then dydx = f′(g(x))·g′(x). Differentiate the outer function, leave the inner function alone, then multiply by the derivative of the inner function.
Differentiate y = ln(x2 + 4x). The outer function is ln u with derivative 1/u. The inner derivative is 2x + 4. So dydx = 2x + 4x2 + 4x.
How does implicit differentiation work?
- Differentiate both sides with respect to x, treating y as a function of x.
- Attach dydx whenever you differentiate a term containing y.
- Collect the dydx terms on one side and solve.
For x2 + y2 = 25, differentiate to get 2x + 2y dydx = 0, so dydx = −xy. At (3, 4) the slope is −3/4.
What are the derivatives of inverse functions?
If g is the inverse of f, then g′(b) = 1/f′(g(b)). The inverse-trigonometric derivatives are:
| Function | Derivative |
|---|---|
| arcsin x | 1/√(1 − x2) |
| arccos x | −1/√(1 − x2) |
| arctan x | 1/(1 + x2) |
Higher-order derivatives repeat differentiation: f″ is the derivative of f′(x), and it measures how fast the slope changes.
Forgetting the inner derivative is the most frequent chain rule error. The derivative of sin(3x) is 3 cos(3x), not cos(3x).
Peel the function like an onion: name the outer layer, differentiate it, then move one layer inward and multiply.
- Chain rule: d/dx f(g(x)) = f′(g(x))·g′(x).
- Implicit differentiation attaches dy/dx to every term containing y.
- The derivative of an inverse at b is the reciprocal of f′ at the matching input.
- Derivatives of arcsin, arccos and arctan come from the inverse function formula.
- The second derivative f″ is the derivative of f′ and measures concavity.
Chain Rule and Implicit Differentiation: frequently asked questions
How do I know when to use the chain rule?
Use it when a function sits inside another function, such as sin(x2) or (3x + 1)5. If you can name an inner and an outer expression, the chain rule applies.
Why does implicit differentiation include dy/dx?
Because y depends on x, so differentiating a power of y needs the chain rule. For example, the derivative of y2 with respect to x is 2y dy/dx.
How do I find a second derivative implicitly?
Differentiate the formula for dy/dx again, using the quotient or product rule. Then substitute the first derivative wherever dy/dx appears and simplify with the original equation.
What does the inverse derivative formula say about slopes?
The tangent slopes of a function and its inverse at matching points are reciprocals. If the slope at (1, 2) is 4, the inverse has slope 1/4 at (2, 1).
Do I need to memorize the inverse trig derivatives?
Yes. The AP exam expects you to recall the derivatives of arcsin x and arctan x, and then apply the chain rule to inputs such as 3x or x2.