Differentiation: Composite, Implicit — AP Calculus AB Unit 3 practice.
This unit covers chain rule, implicit differentiation and inverse functions — essential concepts for AP Calculus AB. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers chain rule, implicit differentiation and inverse functions — essential concepts for AP Calculus AB. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Chain Rule
The chain rule is used to differentiate composite functions of the form f(g(x)). The derivative is f'(g(x)) · g'(x) — the derivative of the outer function evaluated at the inner, multiplied by the derivative of the inner function. This rule applies whenever a function is nested inside another function.
Key Points
- Identify the outer and inner functions before differentiating
- d/dx[f(g(x))] = f'(g(x)) · g'(x) — never forget the inner derivative
- Chain rule chains: if h(x) = f(g(p(x))), differentiate layer by layer from outside in
- Common on the exam with trig, exponential, and logarithmic composites (e.g., sin(x²), e^(3x), ln(x²+1))
Find dy/dx if y = sin(3x² + 1).
The outer function is sin(u) and the inner function is u = 3x² + 1. Differentiating the outer gives cos(u) = cos(3x² + 1), then multiply by the derivative of the inner: 6x. Therefore dy/dx = 6x · cos(3x² + 1).
2 Implicit Differentiation
Implicit differentiation is used when y cannot be easily isolated as a function of x, such as in circle or curve equations. Differentiate both sides with respect to x, applying the chain rule to any y-term by multiplying by dy/dx, then solve algebraically for dy/dx. The exam frequently asks for slope at a given point or a second derivative using this technique.
Key Points
- Every time you differentiate a term involving y, multiply by dy/dx (chain rule)
- After differentiating both sides, collect all dy/dx terms on one side and factor
- To find slope at a point, substitute the given (x, y) values into the expression for dy/dx
- Second derivative problems require substituting dy/dx back into the expression after differentiating again
Given x² + y² = 25, find dy/dx and the slope of the tangent line at (3, 4).
Differentiating both sides with respect to x gives 2x + 2y(dy/dx) = 0. Solving for dy/dx yields dy/dx = −x/y. Substituting the point (3, 4) gives dy/dx = −3/4, which is the slope of the tangent line at that point.
3 Derivatives Of Inverse Functions
If f and g are inverses, then g'(x) = 1 / f'(g(x)). On the exam, you are rarely asked to find the inverse function explicitly; instead, you use a given table or graph to evaluate the derivative of the inverse at a specific point. This formula is also the foundation for the derivatives of arcsin, arccos, and arctan.
Key Points
- Key formula: if g = f⁻¹, then g'(x) = 1 / f'(g(x))
- To use the formula, you need f'(x) and the value of g(x) (the inverse output) at the given point
- Derivatives of inverse trig: d/dx[arctan(x)] = 1/(1+x²), d/dx[arcsin(x)] = 1/√(1−x²)
- Exam tables often give f(a) = b and f'(a) — use these to find (f⁻¹)'(b) = 1/f'(a)
Let f be differentiable and one-to-one. If f(2) = 5 and f'(2) = 3, find (f⁻¹)'(5).
Since f(2) = 5, we know f⁻¹(5) = 2. Applying the inverse function derivative formula: (f⁻¹)'(5) = 1 / f'(f⁻¹(5)) = 1 / f'(2) = 1/3. No algebra or explicit inverse is needed — only the two given values.
Questions, answered.
What is Differentiation: Composite, Implicit?
Differentiation: Composite, Implicit is Unit 3 of AP Calculus AB, covering chain rule, implicit differentiation and inverse functions.
How to study for AP Calculus AB Unit 3?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 30+ review questions across 5 different game modes.