Textbook Question
Consider the curve x=e^y. Use implicit differentiation to verify that dy/dx = e^-y and then find d²y/dx² .
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 3.9.63
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)
Verified step by step guidance1
First, recognize that the function y = 4 log₃(x²−1) can be simplified using the change of base formula for logarithms. The change of base formula is logₐ(b) = ln(b) / ln(a). Therefore, rewrite the function as y = 4 * (ln(x²−1) / ln(3)).
Next, apply the constant multiple rule of differentiation. The derivative of a constant times a function is the constant times the derivative of the function. So, differentiate y = 4 * (ln(x²−1) / ln(3)) with respect to x.
Now, focus on differentiating the natural logarithm part, ln(x²−1). Use the chain rule, which states that the derivative of ln(u) is (1/u) * (du/dx). Here, u = x²−1, so find the derivative of u with respect to x, which is du/dx = 2x.
Substitute the derivative of u back into the chain rule formula. The derivative of ln(x²−1) is (1/(x²−1)) * 2x.
Finally, combine all the parts. The derivative of y = 4 * (ln(x²−1) / ln(3)) is (4/ln(3)) * (2x/(x²−1)). Simplify the expression if necessary.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the curve at any given point. The derivative is denoted as f'(x) or dy/dx and can be calculated using various rules, such as the power rule, product rule, and chain rule.
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Derivatives
Logarithmic Functions
Logarithmic functions are the inverses of exponential functions and are used to simplify complex expressions, especially when dealing with products or powers. The properties of logarithms, such as the product, quotient, and power rules, allow us to rewrite logarithmic expressions in a more manageable form. For example, logₐ(bc) = logₐ(b) + logₐ(c) helps in breaking down the function before differentiation.
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Change of Base Formula
The change of base formula allows us to convert logarithms from one base to another, which can be particularly useful in calculus. For instance, logₐ(b) can be expressed as logₓ(b) / logₓ(a) for any positive x. This is helpful when differentiating logarithmic functions with bases other than e or 10, as it enables the use of natural logarithms, which have simpler derivatives.
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Related Practice
Textbook Question
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = tan¹⁰x / (5x+3)⁶
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
Find the slope of the curve y=sin-1 x at (1/2, π/6) without calculating the derivative of sin-1 x.
Textbook Question
Continuity of a piecewise function Let g(x) = <matrix 2x1> For what values of a is g continuous?