Glossary

3.7 Derivatives of Inverse Functions

2 min read•june 24, 2024

Inverse functions flip the roles of input and output, creating a mirror image of the original function. When we differentiate these flipped functions, we get a reciprocal relationship between their slopes, leading to some nifty formulas for inverse trig functions.

Mastering inverse function derivatives opens up new problem-solving avenues. You'll be able to tackle tricky equations, find slopes of weird curves, and calculate rates of change for complex scenarios. It's a powerful tool in your calculus toolkit.

Derivatives of Inverse Functions

Inverse function theorem application

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  • States if f(x)f(x) is differentiable and one-to-one on an interval, its inverse function f−1(x)f^{-1}(x) is also differentiable
    • Derivative of the inverse function given by ddxf−1(x)=1f′(f−1(x))\frac{d}{dx}f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}
  • Steps to find the derivative of an inverse function:
    1. Given y=f(x)y = f(x), find the inverse function x=f−1(y)x = f^{-1}(y)
    2. Differentiate both sides with respect to yy: dxdy=ddyf−1(y)\frac{dx}{dy} = \frac{d}{dy}f^{-1}(y)
    3. Substitute xx for yy in the derivative to express the result in terms of xx
  • Useful for finding derivatives of functions defined implicitly or as inverses of other functions ()
  • Requires and monotonicity of the original function on its domain

Inverse trigonometric function derivatives

  • Derivatives of :
    • ddxarcsin⁡(x)=11−x2\frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}}
    • ddxarccos⁡(x)=−11−x2\frac{d}{dx}\arccos(x) = -\frac{1}{\sqrt{1-x^2}}
    • ddxarctan⁡(x)=11+x2\frac{d}{dx}\arctan(x) = \frac{1}{1+x^2}
    • ddx\arccot(x)=−11+x2\frac{d}{dx}\arccot(x) = -\frac{1}{1+x^2}
    • ddx\arcsec(x)=1∣x∣x2−1\frac{d}{dx}\arcsec(x) = \frac{1}{|x|\sqrt{x^2-1}}
    • ddx\arccsc(x)=−1∣x∣x2−1\frac{d}{dx}\arccsc(x) = -\frac{1}{|x|\sqrt{x^2-1}}
  • Applications include finding slopes of tangent lines, rates of change, and optimizing functions with inverse trigonometric expressions
    • of the tangent line to y=arctan⁡(x)y = \arctan(x) at x=1x = 1 is 11+12=12\frac{1}{1+1^2} = \frac{1}{2}
    • Rate of change of the angle θ=arcsin⁡(x5)\theta = \arcsin(\frac{x}{5}) with respect to xx when x=3x = 3 is 11−(35)2=54\frac{1}{\sqrt{1-(\frac{3}{5})^2}} = \frac{5}{4}

Slope relationships for inverse functions

  • For a differentiable, one-to-one function f(x)f(x) on an interval:
    • Slopes of tangent lines to f(x)f(x) and f−1(x)f^{-1}(x) at corresponding points are reciprocals
      • If the slope of the tangent line to f(x)f(x) at (a,f(a))(a, f(a)) is mm, the slope of the tangent line to f−1(x)f^{-1}(x) at (f(a),a)(f(a), a) is 1m\frac{1}{m}
  • Derived from the inverse function theorem: ddxf−1(x)=1f′(f−1(x))\frac{d}{dx}f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}
    • Evaluating both sides at x=f(a)x = f(a):
      • ddxf−1(f(a))=1f′(a)\frac{d}{dx}f^{-1}(f(a)) = \frac{1}{f'(a)}
      • Left side is the slope of the tangent line to f−1(x)f^{-1}(x) at (f(a),a)(f(a), a)
      • Right side is the reciprocal of the slope of the tangent line to f(x)f(x) at (a,f(a))(a, f(a))
  • Helps understand the geometric relationship between a function and its inverse (reflection across the line y=xy = x)

Domain, Range, and Derivative Rules

  • The domain of f−1(x)f^{-1}(x) is the range of f(x)f(x), and vice versa
  • Derivative rules for inverse functions build upon basic rules (e.g., chain rule) and are essential for solving complex problems
  • Understanding these concepts helps in analyzing the behavior of inverse functions and their derivatives

Key Terms to Review (12)

Arccos: The inverse cosine function, denoted as arccos or cos^-1, is a trigonometric function that gives the angle whose cosine is equal to a given real number. It is the inverse of the cosine function, allowing one to find the angle given the cosine value.
Arcsin: Arcsin, also known as the inverse sine function, is a mathematical function that allows us to find the angle whose sine is a given value. It is the inverse of the sine function, which maps angles to their corresponding sine values. Arcsin is a crucial concept in the study of inverse functions and their derivatives.
Arctan: Arctan, also known as the inverse tangent function, is a mathematical function that represents the angle whose tangent is a given value. It is a fundamental inverse trigonometric function that is essential in understanding the behavior of inverse functions and their derivatives.
Composition: Composition is the act of combining or arranging multiple elements, functions, or operations into a unified whole. It is a fundamental concept in mathematics and various fields, describing how different components interact and integrate to form a cohesive structure or process.
Continuity: Continuity is a fundamental concept in calculus that describes the smoothness and uninterrupted nature of a function. It is a crucial property that allows for the application of calculus techniques and the study of limits, derivatives, and integrals.
Continuity over an interval: Continuity over an interval means that a function is continuous at every point within a given interval. This implies that the function has no breaks, jumps, or holes in that interval.
F^(-1)(x): The inverse function of a function f(x) is denoted as f^(-1)(x). It represents the function that, when applied to the output of f(x), returns the original input value. The inverse function is a way to 'undo' the original function, allowing us to solve for the input variable when given the output.
Inverse trigonometric functions: Inverse trigonometric functions are the inverse operations of the trigonometric functions (sine, cosine, tangent, etc.), used to find the angle that corresponds to a given trigonometric value. These functions are commonly denoted as $\sin^{-1}(x)$, $\cos^{-1}(x)$, and $\tan^{-1}(x)$.
Inverse Trigonometric Functions: Inverse trigonometric functions are the inverse operations of the basic trigonometric functions (sine, cosine, tangent, etc.). They allow us to find the angle given the value of a trigonometric function, which is essential in various applications of calculus, such as finding derivatives and integrals involving trigonometric functions.
Logarithmic Functions: Logarithmic functions are a class of functions that describe the relationship between two quantities, where one quantity is the exponent that a fixed base must be raised to in order to get the other quantity. They are the inverse functions of exponential functions and have important applications in various fields, including mathematics, science, and engineering.
Point-slope equation: The point-slope equation of a line is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope. It is useful for writing the equation of a line when you know one point and the slope.
Slope: Slope is a measure of the steepness or incline of a line or curve, representing the rate of change between two points. It is a fundamental concept in calculus that underpins the understanding of functions, rates of change, and the shape of graphs.
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