Textbook Question
Find the derivative of the following functions.
y = In x / (In x + 1)
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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 3.10.2
Find the slope of the line tangent to the graph of y = sin^−1 x at x=0.
Verified step by step guidance1
To find the slope of the tangent line to the graph of \( y = \sin^{-1}(x) \) at \( x = 0 \), we need to determine the derivative of \( y \) with respect to \( x \).
The derivative of \( y = \sin^{-1}(x) \) is given by \( \frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}} \). This formula is derived from the inverse trigonometric function differentiation rules.
Substitute \( x = 0 \) into the derivative \( \frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}} \) to find the slope of the tangent line at this point.
Calculate \( \frac{1}{\sqrt{1-0^2}} \), which simplifies to \( \frac{1}{\sqrt{1}} \).
The slope of the tangent line to the graph of \( y = \sin^{-1}(x) \) at \( x = 0 \) is the value obtained from the previous step.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Trigonometric Functions
Inverse trigonometric functions, such as sin^−1(x), are the functions that reverse the action of the standard trigonometric functions. For example, sin^−1(x) gives the angle whose sine is x. Understanding these functions is crucial for evaluating expressions involving them and for finding their derivatives.
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Derivatives of Other Inverse Trigonometric Functions
Derivative of a Function
The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change as the interval approaches zero. For inverse trigonometric functions, knowing how to compute their derivatives is essential for finding slopes of tangents.
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06:30Derivatives of Other Trig Functions
Tangent Line
The tangent line to a curve at a given point is the straight line that just touches the curve at that point, representing the instantaneous rate of change of the function at that point. The slope of the tangent line is given by the derivative of the function evaluated at that point, which is critical for solving problems involving tangents.
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05:13Slopes of Tangent Lines
Related Practice
Textbook Question
Tangent lines Find an equation of the line tangent to the graph of f at the given point.
f(x) = sin−1(x/4); (2,π/6)
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Textbook Question
Derivatives Find and simplify the derivative of the following functions.
s(t) = t⁴/³ / e^t
370
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Textbook Question
63–74. Derivatives of logarithmic functions Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
y = log₈ |tan x|
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Textbook Question
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = (1+x²)^sin x
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Textbook Question
Find the derivative of the following functions.
y = x In x - x
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