Textbook Question
23–51. Calculating derivatives Find the derivative of the following functions.
y = a sin x + b cos x/a sin x - b cos x; a and b are nonzero constants
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 3.9.82
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = x⁸cos³ x / √x-1
Verified step by step guidance1
Step 1: Begin by taking the natural logarithm of both sides of the equation. This will help simplify the differentiation process. Write: ln(f(x)) = ln(x⁸cos³(x)/√(x-1)).
Step 2: Use the properties of logarithms to break down the expression. Apply the logarithm rules: ln(a/b) = ln(a) - ln(b) and ln(a^b) = b*ln(a). This gives: ln(f(x)) = 8ln(x) + 3ln(cos(x)) - (1/2)ln(x-1).
Step 3: Differentiate both sides with respect to x. Remember that the derivative of ln(f(x)) is (1/f(x)) * f'(x). For the right side, differentiate each term separately: d/dx[8ln(x)] = 8/x, d/dx[3ln(cos(x))] = -3sin(x)/cos(x), and d/dx[-(1/2)ln(x-1)] = -(1/2)/(x-1).
Step 4: Combine the derivatives from Step 3 to find the expression for f'(x)/f(x). This results in: f'(x)/f(x) = 8/x - 3tan(x) - 1/(2(x-1)).
Step 5: Solve for f'(x) by multiplying both sides by f(x). This gives: f'(x) = f(x) * (8/x - 3tan(x) - 1/(2(x-1))). Substitute f(x) = x⁸cos³(x)/√(x-1) back into the equation to express f'(x) in terms of x.
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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 complex functions by taking the natural logarithm of both sides. This method simplifies the differentiation process, especially for products, quotients, or powers, by transforming multiplicative relationships into additive ones. It is particularly useful when dealing with functions that involve variable exponents or products of functions.
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Logarithmic Differentiation
Product and Quotient Rules
The product rule and quotient rule are fundamental differentiation rules in calculus. 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 quotient of two functions, ensuring that the derivative accounts for both the numerator and denominator.
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06:43The Quotient Rule
Chain Rule
The chain rule is a crucial differentiation technique used when differentiating composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This rule is essential for handling functions where one function is nested within another, allowing for accurate differentiation of complex expressions.
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Related Practice
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.
g (x) = x^ In x; a = e
Textbook Question
Find the function The following limits represent the slope of a curve y = f(x) at the point (a,f(a)). Determine a possible function f and number a; then calculate the limit.
(lim x🠂2) 1/x+1 - 1/3 / x-2
Textbook Question
Find and simplify the derivative of the following functions.
f(x) = ex(x3 − 3x2 + 6x − 6)
Textbook Question
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) = 1/2x+8; (10,4)
Textbook Question
73–78. {Use of Tech} Normal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the normal line. <IMAGE>
Exercise 46