Unit 03: Differentiation - Composite, Implicit, and Inverse Functions

Question 1

The Chain Rule: Outside/Inside Rule

Answer 1

Definition: A formula for finding the derivative of a composite function f(g(x)). AP Application: Differentiate the ‘outside’ function first (keeping the ‘inside’ exactly as it is), then multiply by the derivative of the ‘inside’ function. Formula: f’(g(x)) * g’(x).

Question 2

Implicit Differentiation: When to Use

Answer 2

Definition: A technique used for equations where y is not isolated (e.g., x^2 + y^2 = 9). AP Application: Differentiate both sides of the equation with respect to x. Every time you differentiate a term containing ‘y’, you must attach a ‘dy/dx’ to it.

Question 3

Steps for Implicit Differentiation

Answer 3

1. Differentiate both sides with respect to x. 2. Collect all terms with ‘dy/dx’ on one side. 3. Factor out ‘dy/dx’. 4. Divide to isolate ‘dy/dx’ to find your slope formula.

Question 4

Higher-Order Implicit Derivatives

Answer 4

To find the second derivative (d^2y/dx^2) implicitly: 1. Differentiate the first derivative. 2. Substitute the original expression for ‘dy/dx’ back into the result. 3. Simplify so the final answer is only in terms of x and y.

Question 5

Inverse Function: Definition and Verification

Answer 5

Definition: A function that reverses the original; if (a, b) is on f, then (b, a) is on the inverse. Verification: Two functions are inverses if f(g(x)) = x and g(f(x)) = x.

Question 6

Derivative of an Inverse Function (Formula)

Answer 6

Formula: The derivative of the inverse g(x) at point ‘a’ is g’(a) = 1 / f’(g(a)). AP Application: This is frequently tested via tables. If you need the derivative of the inverse at x=5, find where the original function’s y-value is 5, then take the reciprocal of the slope at that point.

Question 7

Derivative of Arcsin (Inverse Sine)

Answer 7

Formula: d/dx [arcsin(u)] = u’ / sqrt(1 - u^2). AP Application: Remember the denominator involves a square root and a subtraction sign. Always apply the chain rule (multiply by u’).

Question 8

Derivative of Arctan (Inverse Tangent)

Answer 8

Formula: d/dx [arctan(u)] = u’ / (1 + u^2). AP Application: This is the most common inverse trig derivative on the AP exam. Note that there is no square root in the denominator and the terms are added.

Question 9

Derivative of Arcsec (Inverse Secant)

Answer 9

Formula: d/dx [arcsec(u)] = u’ / (|u| * sqrt(u^2 - 1)). AP Application: Don’t forget the absolute value around the ‘u’ outside the square root.

Question 10

Explicit vs. Implicit Functions

Answer 10

Explicit: y is isolated on one side (y = f(x)). Implicit: x and y are mixed together. In Calculus, implicit functions often represent curves (like circles or ellipses) that are not necessarily functions themselves.