Implicit & Inverse Differentiation
IntermediateImplicit differentiation extends the chain rule to equations not explicitly solved for y, enabling differentiation of curves like circles and ellipses. Inverse function differentiation provides systematic formulas for derivatives of inverse trigonometric, exponential, and logarithmic functions. Together with the chain rule for composite functions and higher-order derivatives, these techniques form the core of AP Calculus AB Unit 3: Differentiation of Composite, Implicit, and Inverse Functions.
The chain rule is the gateway to differentiating composite functions. When equations relate x and y implicitly (like $x^2 + y^2 = 25$), implicit differentiation treats y as a function of x and applies the chain rule to every y-term. Inverse function derivatives extend this further.
Higher-order derivatives reveal concavity, inflection points, and acceleration. Second derivatives found via implicit differentiation often require substituting the first derivative back into the expression.
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- •Apply the chain rule to differentiate composite functions involving trig, exponential, and logarithmic functions
- •Use implicit differentiation to find derivatives of implicitly defined curves
- •Compute derivatives of inverse trigonometric functions using standard formulas and the chain rule
- •Apply logarithmic differentiation to functions with variable bases and exponents
- •Calculate and interpret higher-order derivatives including second and third derivatives