Textbook Question
49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.
h (x) = x^√x; a = 4
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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 3.9.80
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = tan¹⁰x / (5x+3)⁶
Verified step by step guidance1
Step 1: Begin by taking the natural logarithm of both sides of the equation y = (tan^10(x)) / (5x + 3)^6. This gives ln(y) = ln((tan^10(x)) / (5x + 3)^6).
Step 2: Use the properties of logarithms to simplify the expression. The logarithm of a quotient is the difference of the logarithms: ln(y) = ln(tan^10(x)) - ln((5x + 3)^6).
Step 3: Apply the power rule for logarithms, which states that ln(a^b) = b*ln(a). This gives ln(y) = 10*ln(tan(x)) - 6*ln(5x + 3).
Step 4: Differentiate both sides with respect to x. The left side becomes (1/y) * dy/dx by implicit differentiation. The right side requires the chain rule: 10 * (1/tan(x)) * sec^2(x) - 6 * (1/(5x + 3)) * 5.
Step 5: Solve for dy/dx by multiplying both sides by y. Substitute back y = (tan^10(x)) / (5x + 3)^6 to express dy/dx in terms of x. This gives dy/dx = y * [10 * (1/tan(x)) * sec^2(x) - 6 * (1/(5x + 3)) * 5].
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithmic Differentiation
Logarithmic differentiation is a technique used to differentiate functions that are products or quotients of other functions, especially when they involve powers. By taking the natural logarithm of both sides of the function, we can simplify the differentiation process, allowing us to use properties of logarithms to break down complex expressions into manageable parts.
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Logarithmic Differentiation
Chain Rule
The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is essential when dealing with functions raised to powers or nested functions.
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Product and Quotient Rules
The product and quotient rules are essential for differentiating functions that are products or ratios of two or more functions. The product rule states that the derivative of a product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first. The quotient rule, on the other hand, provides a method for differentiating a ratio of two functions, ensuring accurate results when applying these rules in logarithmic differentiation.
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Related Practice
Textbook Question
Consider the curve x=e^y. Use implicit differentiation to verify that dy/dx = e^-y and then find d²y/dx² .
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
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?
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)