Textbook Question
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = tan¹⁰x / (5x+3)⁶
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 3.10.11
Find the slope of the curve y=sin-1 x at (1/2, π/6) without calculating the derivative of sin-1 x.
Verified step by step guidance1
Step 1: Recognize that the slope of the curve at a given point is the derivative of the function at that point. Here, we need the derivative of \( y = \sin^{-1}(x) \).
Step 2: Use the identity \( y = \sin^{-1}(x) \) implies \( \sin(y) = x \). Differentiate both sides with respect to \( x \).
Step 3: Apply implicit differentiation. The derivative of \( \sin(y) \) with respect to \( x \) is \( \cos(y) \cdot \frac{dy}{dx} \), and the derivative of \( x \) is 1.
Step 4: Set up the equation from implicit differentiation: \( \cos(y) \cdot \frac{dy}{dx} = 1 \). Solve for \( \frac{dy}{dx} \) to find \( \frac{dy}{dx} = \frac{1}{\cos(y)} \).
Step 5: Evaluate \( \cos(y) \) at the point \( (1/2, \pi/6) \). Since \( y = \pi/6 \), \( \cos(\pi/6) = \sqrt{3}/2 \). Substitute this into the expression for \( \frac{dy}{dx} \) to find the slope.
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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 reverse the effect of the original function. For example, if y = sin(x), then x = sin<sup>-1</sup>(y) is the inverse function. Understanding how to work with inverse functions is crucial for analyzing their properties, such as slopes and behavior at specific points.
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Inverse Cosine
Slope of a Curve
The slope of a curve at a given point represents the rate of change of the function at that point. It can be interpreted as the tangent line's steepness at the point of interest. For the curve y = sin<sup>-1</sup>(x), finding the slope at (1/2, π/6) involves understanding the relationship between the function and its inverse.
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11:41Summary of Curve Sketching
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. In this context, knowing the values of sin and cos at specific angles, such as π/6, is essential for determining the slope without directly calculating the derivative. These identities help relate the angles to their sine and cosine values.
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7:17Verifying Trig Equations as Identities
Related Practice
Textbook Question
Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas y = cx² form orthogonal trajectories with the family of ellipses x²+2y² = k, where c and k are constants (see figure).
Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. <IMAGE>
y = cx²; x²+2y² = k, where c and k are constants
Textbook Question
Derivatives Find and simplify the derivative of the following functions.
f(x) = x /x+1
Textbook Question
15–48. Derivatives Find the derivative of the following functions.
y = 10^x(In 10^x-1)
Textbook Question
Derivatives Find and simplify the derivative of the following functions.
g(t) = 3t² + 6/t⁷
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Textbook Question
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 = 4 log₃(x²−1)