Derivatives of Composite, Implicit, and Inverse Functions
Lessons in this chapter
About Derivatives of Composite, Implicit, and Inverse Functions
Derivatives of Composite, Implicit, and Inverse Functions extends differentiation to every remaining function type and structure.
Differentiating Composite Functions introduces the chain rule for differentiating one function nested inside another, and Differentiating Exponential, Logarithmic, and Inverse Functions applies it (and other rules) to those specific function families.
Implicit and Logarithmic Differentiation covers differentiating an equation that isn’t solved for y, and using logarithms to simplify differentiating a complicated product or power, while Differentiating Inverse Trigonometric Functions gives the specific derivative rules for each inverse trig function.
Finding Higher-Order Derivatives closes the chapter by differentiating a function more than once, to find its rate of change’s own rate of change.
The five lessons, from Differentiating Composite Functions to Finding Higher-Order Derivatives, each open as a practice worksheet in the page. Problems are scored as you answer them, so a chain rule applied one layer short is caught at that question. Free to practice online.
Frequently asked questions
Differentiating Composite Functions is the chain rule, and the rest of the chapter leans on it. Differentiating Exponential, Logarithmic, and Inverse Functions extends the basic rules, Implicit and Logarithmic Differentiation handles equations that are not solved for y and products that are easier to differentiate after taking logs, Differentiating Inverse Trigonometric Functions adds those formulas, and Finding Higher-Order Derivatives repeats the process to reach the second derivative and beyond. This is Unit 3 of the AP Calculus AB course framework, Differentiation: Composite, Implicit, and Inverse Functions, well beyond the high school Common Core.
The chain rule differentiates a composite function by multiplying the outer function’s derivative (evaluated at the inner function) by the inner function’s own derivative.
Implicit differentiation differentiates both sides of an equation that isn’t solved for y, treating y as a function of x and applying the chain rule wherever y appears.
Logarithmic differentiation takes the logarithm of both sides before differentiating, turning a complicated product, quotient, or power into a much simpler sum or difference first.
Each inverse trig function has its own specific derivative rule, derived using implicit differentiation on the original trig function’s equation.
A higher-order derivative is found by differentiating a function’s derivative again (and again), giving the rate of change of the rate of change, and so on.