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Chapter 3: Problem 16

Find the following derivatives. $$\frac{d}{d x}\left(e^{x} \ln x\right)$$

Short Answer

Expert verified
Question: Find the derivative of the function \(e^x \ln x\) with respect to \(x\). Answer: The derivative of the function \(e^x \ln x\) with respect to \(x\) is \(\frac{d}{d x}\left(e^{x} \ln x\right) = e^x \cdot \left(\ln x + \frac{1}{x} \right)\).

Step by step solution

01

Identify the two functions

In this case, the given function is \(e^x \ln x\), which is a product of two functions, \(f(x)=e^x\) and \(g(x)=\ln x\). We will find the derivatives of these two functions and then apply the product rule.
02

Find the derivatives of the two functions

Now, let's find the derivatives of the two functions: 1. Derivative of \(f(x)=e^x\) is \(f'(x)=e^x\). 2. Derivative of \(g(x)=\ln x\) is \(g'(x)=\frac{1}{x}\).
03

Apply the product rule

According to the product rule, the derivative of a product of two functions is given by: $$(f(x) \cdot g(x))' = f'(x) \cdot g(x) + f(x) \cdot g'(x)$$ Now, we substitute the functions and their derivatives: $$\left(e^x \cdot \ln x\right)' = e^x \cdot \ln x + e^x \cdot \frac{1}{x}$$ So the derivative is: $$\frac{d}{d x}\left(e^{x} \ln x\right) = e^x \cdot \left(\ln x + \frac{1}{x} \right)$$

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Most popular questions from this chapter

The Witch of Agnesi The graph of \(y=\frac{a^{3}}{x^{2}+a^{2}},\) where \(a\) is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi). a. Let \(a=3\) and find an equation of the line tangent to \(y=\frac{27}{x^{2}+9}\) at \(x=2\) b. Plot the function and the tangent line found in part (a).

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Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}\left[\frac{x f(x)}{g(x)}\right]\right|_{x=4}$$
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