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

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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