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Chapter 5: Problem 47

Find the derivative.\(y=\ln e^{x}\)

Short Answer

Expert verified
The derivative of the function \(y=\ln e^{x}\) is 1.

Step by step solution

01

Recognize the function

First recognize the function \(y=\ln e^{x}\). Here, \(e^{x}\) is a function of \(x\) inside the logarithm function, making it a composite function.
02

Substitute and Simplify

However, since \(e\) is the base of the natural logarithm, we have that \(\ln(e^{x}) = x\). So the function simplifies to \(y=x\).
03

Differentiate

The derivative of \(y=x\) with respect to \(x\) is simply 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithms
Natural logarithms are special because they use the number \(e\) as their base, and \(e\) is approximately 2.71828. This base is used frequently in mathematics, particularly when dealing with growth and decay processes. One of the important properties to know about natural logarithms is that the logarithm of \(e\) to any power simplifies easily.
  • For any real number \(x\), the natural logarithm \(\ln(e^x) = x\).
  • This property arises because logarithms are the inverse operations of exponentiation.
  • Hence, whenever we take the natural log of an exponential expression, where \(e\) is the base, it simplifies directly to the exponent of \(e\).
Understanding this train of thought makes it easier to work with natural logarithms and decode complicated expressions involving them.
Composite Functions
In mathematics, a composite function is a function made up of two or more simpler functions. When dealing with functions like \(y = \ln(e^x)\), you're looking at a composition of two functions: the natural logarithm function and the exponential function.
  • The outer function here is \(\ln\), representing the natural logarithm.
  • The inner function is \(e^x\), which is an exponential function.
Composite functions are essential because they help represent complicated relationships more simply. Often, in calculus, we differentiate composite functions using techniques like the chain rule. Recognizing composite functions lays the groundwork to apply these rules effectively.
Chain Rule
The chain rule is a fundamental rule in calculus for finding the derivative of composite functions. While it wasn't needed for our original problem (since simplifying \(\ln(e^x)\) to \(x\) made the derivative straightforward), understanding it is crucial for more complex scenarios.
  • The chain rule states that if you have a function \(y = f(g(x))\), the derivative, \(y'\), is \(f'(g(x)) \cdot g'(x)\).
  • Essentially, you differentiate the outer function and multiply by the derivative of the inner function.
This rule is especially helpful when faced with less apparent simplifications in composite functions. Mastering the chain rule empowers you to tackle various mathematical challenges with greater confidence.

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Most popular questions from this chapter

Evaluate the integral. $$ \int_{0}^{4} \frac{1}{\sqrt{25-x^{2}}} d x $$

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Use the functions \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the given function. \((f \circ g)^{-1}\)

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Let \(x>0\) and \(b>0 .\) Show that \(\int_{-b}^{b} e^{x t} d t=\frac{2 \sinh b x}{x}\).
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