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Chapter 6: Problem 74

If \(y=\left(1+\frac{1}{x}\right)^{x}+x^{\left(1+\frac{1}{x}\right)}\), find \(\frac{d y}{d x}\) at \(x=1\)

Short Answer

Expert verified
The value of \(\frac{d y}{d x}\) at \(x=1\) is 4.

Step by step solution

01

Differentiate the function

First, the function must be differentiated. It is a sum, so differentiate each term separately. The first term \(\left(1+\frac{1}{x}\right)^{x}\) can be differentiated using the chain rule. To differentiate this term, replace \(x\) with \(\log(x)\), differentiate the term, and then substitute \(\log(x)\) back with \(x\). This gives us \(-x*\left(1+\frac{1}{x}\right)^{x-1}\).\n\n For the second part \(x^{\left(1+\frac{1}{x}\right)}\), apply logarithmic differentiation, i.e., take logarithms of both sides, differentiate and then substitute back. Doing this simplifies the term to \(1*\left(1+\frac{1}{x}\right)*x^{\left(1+\frac{1}{x}\right)-1}\).
02

Simplify and Combine

Now, it's time to simplify and combine the terms from both the parts of the function. This gives the entire derivative of the function as \(\frac{d y}{d x}=-x*\left(1+\frac{1}{x}\right)^{x-1}+1*\left(1+\frac{1}{x}\right)*x^{\left(1+\frac{1}{x}\right)-1}\).
03

Evaluate the derivative at the point x=1

Now, plug in the given point \(x=1\) into the derivative function. This gives \(\frac{d y}{d x}\) at \(x=1\) as \(2*(2^{0}+1^{(2-1)})=4\).

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