Problem 5 Differentiate the following func... [FREE SOLUTION] (2024)

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When differentiating a product of two functions, we use the Product Rule. The Product Rule states that the derivative of a product of two functions \(f(x)\) and \(g(x)\) is given by:

d(fg)/dx = f'(x)g(x) + f(x)g'(x)

This means we differentiate the first function and multiply it by the second function, then add the product of the first function and the derivative of the second function. This rule ensures that we account for the rate of change in both functions.
In the provided exercise, we identified \(f(x) = e^x\) and \(g(x) = \text{ln} \, x\). Their derivatives are \(f'(x) = e^x\) and \(g'(x) = 1/x\), respectively.
Applying the Product Rule, we combined the derivatives to get:

y' = f'(x) \, g(x) + f(x) \, g'(x)
Substituting in the known derivatives results in: \(e^x \, \text{ln} \, x + e^x \, \frac{1}{x}\).

The natural logarithm, denoted as \(\text{ln} \, x\), is a logarithm with the base \(e\), where \(e\) is approximately equal to 2.71828. The natural logarithm has several important properties:

• When \(x = 1\), \(\text{ln} \, 1 = 0\)
• It is the inverse function of the exponential function \(e^x\)
• The derivative of \(\text{ln} \, x\) with respect to \(x\) is \(\frac{1}{x}\)

In the exercise, one of the functions we differentiated was \(\text{ln} \, x\). Understanding how to handle natural logarithms and their properties is essential in differentiation, especially when combined with other types of functions.

Exponential functions are functions of the form \(e^x\), where \(e\) is Euler's number. These functions grow very quickly and have unique differentiation properties:

• The derivative of \(e^x\) is still \(e^x\)
• They play a crucial role in describing growth and decay processes in natural and social sciences

In our differentiation exercise, the function \(e^x\) is an integral part of the original function \(y = e^x \, \text{ln} \, x\). Because the derivative of \(e^x\) is itself, it simplifies the process of using the Product Rule.
Consequently, when we combined the derivatives using the Product Rule, the resulting expression was simplified as: \(e^x \,(\text{ln} \, x + \frac{1}{x})\).This shows the power and simplicity of exponential functions in calculus.