Textbook Question
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = (1+x²)^sin x
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6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Logarithmic Differentiation
5:50 minutesProblem 7.1.25
Textbook Question
Evaluate the following derivatives.
d/dx ((1/x)ˣ)
Verified step by step guidance1
Step 1: Recognize that the function is of the form \( f(x) = \left( \frac{1}{x} \right)^x \), which involves both a base and an exponent that depend on \( x \). This requires the use of logarithmic differentiation.
Step 2: Take the natural logarithm of both sides to simplify the expression. Let \( y = \left( \frac{1}{x} \right)^x \), then \( \ln(y) = x \ln\left( \frac{1}{x} \right) \).
Step 3: Simplify \( \ln\left( \frac{1}{x} \right) \) using logarithmic properties: \( \ln\left( \frac{1}{x} \right) = \ln(1) - \ln(x) = -\ln(x) \). Thus, \( \ln(y) = x(-\ln(x)) = -x \ln(x) \).
Step 4: Differentiate both sides with respect to \( x \). Using implicit differentiation, \( \frac{d}{dx}[\ln(y)] = \frac{1}{y} \frac{dy}{dx} \) and \( \frac{d}{dx}[-x \ln(x)] = -\ln(x) - x \cdot \frac{1}{x} \).
Step 5: Solve for \( \frac{dy}{dx} \) by multiplying through by \( y \). Substitute \( y = \left( \frac{1}{x} \right)^x \) back into the equation to express the derivative 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.
Derivative
A derivative represents the rate at which a function changes at a given point. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. The derivative is often denoted as f'(x) or df/dx, and it can be interpreted as the slope of the tangent line to the function's graph at a specific point.
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Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a constant and 'x' is the variable. In the context of derivatives, exponential functions have unique properties, particularly when the base is the natural number 'e'. The derivative of an exponential function is proportional to the function itself, which simplifies the differentiation process.
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Logarithmic Differentiation
Logarithmic differentiation is a technique used to differentiate functions that are products or quotients of variables raised to variable powers. By taking the natural logarithm of both sides of the equation, one can simplify the differentiation process, especially for complex functions like (1/x)ˣ. This method is particularly useful when dealing with functions where the variable appears in both the base and the exponent.
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