Textbook Question
Find d/dx(ln√x²+1).
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.10.51
47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.
f(x)=tan x; (1,π/4)
Verified step by step guidance1
Identify the function given: \( f(x) = \tan x \). We need to find the derivative of its inverse at the point \((1, \pi/4)\).
Recall that if \( y = f^{-1}(x) \), then \( f(y) = x \). The derivative of the inverse function at a point \( x = a \) is given by \( (f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))} \).
Since \( f(x) = \tan x \), the derivative \( f'(x) = \sec^2 x \).
We know \( f(\pi/4) = \tan(\pi/4) = 1 \), so \( f^{-1}(1) = \pi/4 \).
Substitute \( f^{-1}(1) = \pi/4 \) into the formula for the derivative of the inverse: \( (f^{-1})'(1) = \frac{1}{f'(\pi/4)} = \frac{1}{\sec^2(\pi/4)} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Functions
Inverse functions are functions that 'reverse' the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f⁻¹(y) takes y back to x. Understanding how to find and work with inverse functions is crucial for evaluating derivatives of inverses.
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Inverse Cosine
Derivative of Inverse Functions
The derivative of an inverse function can be calculated using the formula (f⁻¹)'(y) = 1 / f'(x), where y = f(x). This relationship shows that the slope of the tangent line to the inverse function at a point is the reciprocal of the slope of the tangent line to the original function at the corresponding point.
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Trigonometric Functions and Their Derivatives
Trigonometric functions, such as f(x) = tan(x), have specific derivatives that are essential for solving problems involving these functions. For example, the derivative of tan(x) is sec²(x). Knowing these derivatives allows for the application of the inverse derivative formula effectively in the context of trigonometric functions.
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6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
√x⁴+y² = 5x+2y³
Textbook Question
27–76. Calculate the derivative of the following functions.
Textbook Question
9–61. Evaluate and simplify y'.
y = (2x−3)x^3/2
Textbook Question
Find an equation of the line tangent to the curve y = sin x at x = 0.
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Textbook Question
A 12-ft ladder is leaning against a vertical wall when Jack begins pulling the foot of the ladder away from the wall at a rate of 0.2 ft/s. What is the configuration of the ladder at the instant when the vertical speed of the top of the ladder equals the horizontal speed of the foot of the ladder?