Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
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.
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