Glossary

9.1 Derivatives of inverse functions

2 min read•july 22, 2024

Inverse functions flip inputs and outputs, allowing us to "undo" operations. They're like mathematical mirrors, reflecting functions across y=x. Understanding their derivatives is crucial for solving complex problems in calculus.

Knowing how to find derivatives of inverse functions opens doors to tackling optimization and related rates problems. It's a powerful tool that lets us analyze relationships between variables in reverse, expanding our problem-solving toolkit.

Inverse Functions and Their Derivatives

Concept of inverse functions

Top images from around the web for Concept of inverse functions

  • Use the graph of a function to graph its inverse | College Algebra View original

    Is this image relevant?

  • Inverse Functions · Algebra and Trigonometry View original

    Is this image relevant?

  • Use the graph of a function to graph its inverse | College Algebra View original

    Is this image relevant?

1 of 3

Top images from around the web for Concept of inverse functions

  • Use the graph of a function to graph its inverse | College Algebra View original

    Is this image relevant?

  • Inverse Functions · Algebra and Trigonometry View original

    Is this image relevant?

  • Use the graph of a function to graph its inverse | College Algebra View original

    Is this image relevant?

1 of 3
  • Definition states that if f(x)f(x) and g(x)g(x) are inverse functions, then composing them in either order yields the original input (f(g(x))=g(f(x))=xf(g(x)) = g(f(x)) = x)
  • Graphically, the inverse function is a reflection of the original function across the line y=xy = x (swaps xx and yy coordinates of each point)
  • Relationship between derivatives: if f(x)f(x) and g(x)g(x) are inverses and both differentiable, then g′(x)=1f′(g(x))g'(x) = \frac{1}{f'(g(x))} (reciprocal of derivative evaluated at inverse)
    • Derived using chain rule and fact that f(g(x))=xf(g(x)) = x (composing inverses yields identity function)

Formula for inverse derivatives

  • If y=[f−1(x)](https://www.fiveableKeyTerm:f−1(x))y = [f^{-1}(x)](https://www.fiveableKeyTerm:f^{-1}(x)) is the inverse of x=f(y)x = f(y), then dydx=1f′(y)\frac{dy}{dx} = \frac{1}{f'(y)} (derivative of inverse equals reciprocal of derivative of original function)
  • Steps to find derivative of inverse:
    1. Given y=f−1(x)y = f^{-1}(x), write x=f(y)x = f(y) (swap xx and yy to get original function)
    2. Find dxdy\frac{dx}{dy} using derivative of original function (differentiate with respect to yy)
    3. Solve for dydx\frac{dy}{dx} by taking reciprocal of dxdy\frac{dx}{dy} (invert to get derivative of inverse)
    4. Replace yy with f−1(x)f^{-1}(x) in resulting expression (substitute inverse function for yy)

Derivatives of inverse trigonometric functions

  • ddxarcsin⁡(x)=11−x2\frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1-x^2}} (derivative of arcsine)
  • ddxarccos⁡(x)=−11−x2\frac{d}{dx} \arccos(x) = -\frac{1}{\sqrt{1-x^2}} (derivative of arccosine, negative due to decreasing function)
  • ddxarctan⁡(x)=11+x2\frac{d}{dx} \arctan(x) = \frac{1}{1+x^2} (derivative of arctangent)
  • ddx\arccot(x)=−11+x2\frac{d}{dx} \arccot(x) = -\frac{1}{1+x^2} (derivative of arccotangent, negative due to decreasing function)
  • ddx\arcsec(x)=1∣x∣x2−1\frac{d}{dx} \arcsec(x) = \frac{1}{|x|\sqrt{x^2-1}} (derivative of arcsecant, absolute value since secant is always positive)
  • ddx\arccsc(x)=−1∣x∣x2−1\frac{d}{dx} \arccsc(x) = -\frac{1}{|x|\sqrt{x^2-1}} (derivative of arccosecant, negative due to decreasing function)
  • Domain of inverse function equals range of original function (restricts inputs to ensure one-to-one correspondence)
    • arcsin⁡(x)\arcsin(x) has domain [−1,1][-1, 1] since range of sin⁡(x)\sin(x) is [−1,1][-1, 1] (sine outputs between -1 and 1)

Applications of inverse function derivatives

  • Optimization problems involve finding maximum or minimum values (critical points where derivative equals zero)
  • Related rates problems require applying chain rule to derivatives of inverse functions (rates of change depend on each other)
  • Strategies:
    1. Identify inverse function and its derivative (recognize function composition)
    2. Apply chain rule when necessary (multiple variables changing with respect to time or each other)
    3. Use domain restrictions to check validity of solution (ensure inputs are within allowed range)

Key Terms to Review (21)

(f^-1)'(x) = 1/(f'(f^-1(x))): This equation represents the derivative of an inverse function, showing that the rate of change of the inverse function at a point x can be found by taking the reciprocal of the rate of change of the original function at the corresponding point. The formula highlights how the slopes of inverse functions are related, allowing for the calculation of derivatives without directly differentiating the inverse itself. Understanding this relationship is crucial for analyzing functions and their inverses in calculus.
Analyzing concavity: Analyzing concavity involves determining the direction in which a function curves, specifically whether it is concave up (curving upwards) or concave down (curving downwards). This concept is crucial for understanding the behavior of functions and their graphical representations, as it relates to the second derivative of the function and can indicate points of inflection where the concavity changes.
Arccos(x): The arccosine function, denoted as arccos(x), is the inverse function of the cosine function, returning the angle whose cosine is x. This means if $$y = ext{arccos}(x)$$, then $$x = ext{cos}(y)$$ for angles $$y$$ in the range from 0 to $$ rac{ ext{pi}}{2}$$ radians. The arccos function is crucial for understanding how inverse functions work, particularly in finding angles from given cosine values.
Arccot(x): The function arccot(x) is the inverse of the cotangent function, which gives the angle whose cotangent is x. This means if y = arccot(x), then cot(y) = x. This function is important in calculus because it helps in finding angles associated with cotangent values, and it plays a crucial role when dealing with derivatives of inverse functions.
Arccsc(x): The arccsc(x), or inverse cosecant function, is the function that returns the angle whose cosecant is x. It is defined as the inverse of the cosecant function, which is itself the reciprocal of the sine function. This function helps us find angles when given a specific value of cosecant and plays a significant role in understanding inverse functions and their derivatives.
Arcsec(x): The function arcsec(x) is the inverse of the secant function, defined for values of x where |x| ≥ 1. It gives the angle whose secant is x, allowing you to find an angle in a right triangle when given the length of the hypotenuse and the adjacent side. Understanding arcsec(x) is crucial because it highlights how inverse trigonometric functions work and how they relate to their original functions.
Arcsin(x): arcsin(x) is the inverse function of the sine function, specifically defined for input values in the range of -1 to 1. It returns the angle whose sine is the given value, producing results in radians from -$$\frac{\pi}{2}$$ to $$\frac{\pi}{2}$$. Understanding arcsin(x) is crucial when studying inverse functions, as it showcases how the behavior of a function can be reversed to find the original angle from a given sine value.
Arctan(x): The function arctan(x) is the inverse of the tangent function, which returns the angle whose tangent is x. This means that if you have a value for the tangent, you can use arctan to find the angle in radians. The output of arctan(x) is restricted to the interval (-π/2, π/2), ensuring a unique output for each input value.
Augustin-Louis Cauchy: Augustin-Louis Cauchy was a French mathematician who made significant contributions to analysis and calculus, particularly in the development of the concept of limits, continuity, and differentiability. His work laid the groundwork for many modern mathematical theories, particularly in understanding inverse functions, the Mean Value Theorem, and the application of L'Hôpital's Rule in solving indeterminate forms.
Continuity: Continuity in mathematics refers to a property of a function where it does not have any breaks, jumps, or holes over its domain. A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This concept is crucial because it ensures that the behavior of functions can be analyzed smoothly, impacting several important mathematical principles and theorems.
Derivative of the inverse function: The derivative of the inverse function is a mathematical concept that describes how the rate of change of a function relates to the rate of change of its inverse. Specifically, if a function $$f$$ is continuous and has a non-zero derivative at a point, then its inverse $$f^{-1}$$ is also differentiable at the corresponding point, and the relationship between their derivatives is given by the formula: $$\frac{d}{dx}(f^{-1}(y)) = \frac{1}{f'(x)}$$ where $$y = f(x)$$. This relationship highlights how understanding the original function's behavior can provide insights into its inverse's behavior.
Differentiability: Differentiability refers to the property of a function being differentiable at a point or on an interval, which means it has a defined derivative at that point or throughout that interval. This concept is essential in understanding how functions behave, as it indicates smoothness and continuity, allowing for the application of various calculus principles. Differentiability also plays a crucial role in analyzing inverse functions, exponential functions, critical points, limits, and iterative methods for finding roots of equations.
Dy/dx: The notation $$\frac{dy}{dx}$$ represents the derivative of a function, indicating the rate at which the dependent variable $$y$$ changes with respect to the independent variable $$x$$. This concept is essential for understanding how functions behave and helps in solving problems related to tangents, slopes, and rates of change. The derivative encapsulates the instantaneous rate of change, allowing for the analysis of motion and the dynamics of systems.
F^{-1}(x): The notation f^{-1}(x) represents the inverse function of f(x), which essentially reverses the effect of the original function. When you apply f to a value and then apply f^{-1} to the result, you get back to the original input. Inverse functions are crucial for understanding how functions behave and how their derivatives relate to each other, especially in calculus.
Finding slopes of tangent lines: Finding slopes of tangent lines refers to the process of determining the steepness or incline of a curve at a specific point. This involves using the concept of derivatives, where the derivative of a function at a point gives the slope of the tangent line to the curve at that point. This idea is crucial in understanding how functions behave locally and is especially relevant when working with inverse functions and logarithmic differentiation, where these slopes help analyze relationships between variables.
Gottfried Wilhelm Leibniz: Gottfried Wilhelm Leibniz was a German mathematician and philosopher, best known for co-developing calculus independently of Isaac Newton. His work laid the foundations for many concepts in mathematics, including the notation used for derivatives and integrals. Leibniz's contributions are significant in understanding inverse functions and the applications of antiderivatives, linking his theories to essential principles in calculus.
Horizontal Line Test: The horizontal line test is a method used to determine if a function is one-to-one, which means that each output corresponds to exactly one input. If any horizontal line intersects the graph of the function more than once, the function fails the test and is not one-to-one. This concept is crucial in understanding how functions behave, especially when discussing inverse functions and their properties.
Inverse Function Theorem: The Inverse Function Theorem states that if a function is continuously differentiable and its derivative is non-zero at a point, then there exists a neighborhood around that point where the function has a continuous inverse. This theorem is essential in understanding the relationship between functions and their inverses, particularly in how to derive the derivatives of these inverses in various contexts.
Ln(x) and e^x: The natural logarithm function, denoted as ln(x), is the inverse of the exponential function e^x. This means that if y = e^x, then x = ln(y). Both functions are crucial in calculus because they exhibit unique properties, such as their derivatives being closely linked to one another and their ability to model continuous growth processes.
One-to-One Function: A one-to-one function is a type of function where each input corresponds to exactly one unique output, and no two different inputs produce the same output. This property ensures that the function has an inverse that is also a function, allowing for the relationship between the input and output to be reversed without ambiguity. Being one-to-one is crucial in understanding function composition, defining inverses accurately, and analyzing derivatives of inverse functions.
Sin(x) and arcsin(x): The sine function, denoted as sin(x), is a fundamental trigonometric function that represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. The arcsine function, denoted as arcsin(x), is the inverse of the sine function, providing the angle whose sine is a given value. These functions are interconnected, as understanding their derivatives and relationships is crucial for comprehending inverse functions.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide