In Exercises 59–86, find the derivative of y with respect to the given independent variable.
81. y = log₁₀(e^x)

검증된 단계별 안내

Recognize that the function is given as \(y = \log_{10}(e^x)\), which is a logarithm with base 10 of the expression \(e^x\).

Use the logarithm property that allows you to rewrite \(\log_{10}(e^x)\) as \(x \cdot \log_{10}(e)\), since \(\log_b(a^c) = c \cdot \log_b(a)\).

Recall that \(\log_{10}(e)\) is a constant because it does not depend on \(x\).

Differentiate \(y = x \cdot \log_{10}(e)\) with respect to \(x\) using the constant multiple rule, which states that the derivative of \(c \cdot f(x)\) is \(c \cdot f'(x)\) where \(c\) is a constant.

Since the derivative of \(x\) with respect to \(x\) is 1, the derivative \(\frac{dy}{dx}\) simplifies to \(\log_{10}(e)\).

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주요 개념

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Logarithmic Functions and Their Properties

Logarithmic functions are the inverses of exponential functions. Understanding properties like logₐ(b^x) = x·logₐ(b) helps simplify expressions before differentiation. In this problem, recognizing that log₁₀(e^x) can be rewritten using these properties is essential.

Derivative of Exponential Functions

The derivative of an exponential function e^x with respect to x is e^x. This fundamental rule is crucial when differentiating expressions involving exponentials, especially when combined with logarithmic functions.

Chain Rule for Differentiation

The chain rule is used to differentiate composite functions. When a function is nested inside another, like log₁₀(e^x), the derivative is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

In Exercises 59–86, find the derivative of y with respect to the given independent variable.81. y = log₁₀(e^x)