Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.9.94

Chapter 3, Problem 3.9.94

Calculate the derivative of the following functions (i) using the fact that b^x = e^{xIn b} and (ii) using logarithmic differentiation. Verify that both answers are the same.
y = (4x+1)^{In x}

Verified step by step guidance

Step 1: To find the derivative using the fact that b^x = e^(x ln b), rewrite the function y = (4x+1) ln x in terms of exponential functions. This involves expressing any power terms in the form of e^(x ln b).

Step 2: Apply the product rule to differentiate y = (4x+1) ln x. The product rule states that if you have a function y = u*v, then the derivative y' = u'v + uv'. Here, u = (4x+1) and v = ln x.

Step 3: Differentiate u = (4x+1) to get u' = 4. Differentiate v = ln x to get v' = 1/x. Substitute these derivatives into the product rule formula to find the derivative of y.

Step 4: For logarithmic differentiation, take the natural logarithm of both sides of the equation y = (4x+1) ln x, which gives ln y = ln((4x+1) ln x). Differentiate both sides with respect to x using implicit differentiation.

Step 5: After differentiating, solve for dy/dx. Compare the derivative obtained from logarithmic differentiation with the derivative obtained using the product rule to verify that both methods yield the same result.

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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 of the function at any given point. The derivative can be calculated using various rules, such as the power rule, product rule, and chain rule, depending on the form of the function.

Exponential and Logarithmic Functions

Exponential functions are of the form b^x, where b is a constant and x is the variable. They can be expressed using natural logarithms as b^x = e^(x ln b). Logarithmic functions, the inverses of exponential functions, help simplify the differentiation of complex expressions, especially when dealing with products or powers, making them essential for techniques like logarithmic differentiation.

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products or powers of variables. By taking the natural logarithm of both sides of the equation, the differentiation process becomes simpler, allowing the use of properties of logarithms to break down complex expressions. This method is particularly useful for functions where the variable is in both the base and the exponent.

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Logarithmic Differentiation

Related Practice

Textbook Question

Find (f^−1)′(3), where f(x)=x³+x+1.

Textbook Question

Suppose f is a one-to-one function with f(2)=8 and f′(2)=4. What is the value of (f^−1)′(8)?

Textbook Question

Use implicit differentiation to find dy/dx.

sin xy = x+y

Textbook Question

90–93. {Use of Tech} Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure that they actually lie on the curve. Confirm your results with a graph.

x(1−y²)+y³=0

Textbook Question

Derivatives Find and simplify the derivative of the following functions.

g(x) = e^x / x²-1

Textbook Question

A challenging second derivative Find d²y/dx², where √y+xy=1.

Calculate the derivative of the following functions (i) using the fact that bx = exIn b and (ii) using logarithmic differentiation. Verify that both answers are the same.y = (4x+1)In x

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Key Concepts

Derivative

Exponential and Logarithmic Functions

Logarithmic Differentiation

Calculate the derivative of the following functions (i) using the fact that b^x = e^{xIn b} and (ii) using logarithmic differentiation. Verify that both answers are the same.
y = (4x+1)^{In x}