Topic 3.3-3.4: Derivatives of Inverse Functions and Inverse Trigonometric Functions

Inverse Function

A function that reverses the effect of the original function; denoted as f^{-1}(x).

Reciprocal Slope Relationship

The slopes of a function and its inverse at corresponding points are reciprocals of each other.

Inverse Function Derivative Theorem

States that if f has an inverse g, then g'(x) = 1/f'(g(x)) for all x where f'(g(x)) ≠ 0.

Tangent Line Reflection

The tangent lines at corresponding points of a function and its inverse are reflections across the line y=x.

Differentiable Function

A function that has a derivative at every point in its domain.

Slope of the Inverse

To find the slope of the inverse, evaluate the reciprocal of the derivative of the original function at the corresponding point.

Step 1 in Finding (f^{-1})'(b)

Identify the goal: the slope of the inverse at x = b.

Step 2 in Finding (f^{-1})'(b)

Find the corresponding coordinate by solving f(a) = b.

Step 3 in Finding (f^{-1})'(b)

Find the derivative of the original function by calculating f'(a).

Step 4 in Finding (f^{-1})'(b)

Reciprocate to obtain the slope of the inverse: 1/f'(a).

Arcsin Derivative Formula

The derivative of arcsin(u) is u' / √(1-u²).

Arccos Derivative Formula

The derivative of arccos(u) is -u' / √(1-u²).

Arctan Derivative Formula

The derivative of arctan(u) is u' / (1+u²).

Arcsec Derivative Formula

The derivative of arcsec(u) is u' / |u|√(u²-1).

Common Mistake: Notation Confusion

Mistaking f^{-1}(x) for 1/f(x) or sin^{-1}(x) for 1/sin(x).

Common Mistake: Evaluating at Wrong Value

Calculating (f^{-1})'(5) at the incorrect value; must find x where f(x)=5.

Common Mistake: Chain Rule Omission

Failing to apply the chain rule when differentiating inverse trig functions.

Common Mistake: Forgetting Absolute Values

Omitting absolute values in the derivatives of arcsec and arccsc.

Geometric Reflection

The graph of an inverse function is the geometric reflection of the original function across the line y=x.

Differentiability Condition

g is differentiable at x if f'(g(x)) ≠ 0.

Coordinate Swap

When evaluating derivatives of inverse functions, switch the input and output coordinates.

Output of Original Function

The value calculated from the function f that corresponds to the inverse input.

Worked Example

A demonstration of applying the theorem, including setting up and solving for derivatives.

AP Exam Focus

Inverse trigonometric derivatives are commonly tested on the AP Calculus AB exam.

Memory Aids for Derivatives

'Co' rule for negative derivatives of inverse trig functions, etc.

Trigonometric Identity for Derivatives

Sine and cosine involve 1, while secant and cosecant involve u in their derivatives.

Tangent and Cotangent Note

Tangent and cotangent are unique for having no square root in their derivative formulas.

Inverse Trigonometric Function Notation

Understanding that arcsin x is the same as sin^{-1} x and arctan x is the same as tan^{-1} x.