Calculating Derivative – A function to the power of a function – Exercise 6377

Exercise

Find the derivative of the following function:

$f(x)={(1+\frac{1}{x})}^x$

Final Answer

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Solution

We do not have a derivative formula for a function to the power of a function. To work around this, we use a “trick” – we use logarithm rules to get a multiplication of functions instead of a function to the power of a function.

$f(x)={(1+\frac{1}{x})}^x=$

$=e^{\ln {(1+\frac{1}{x})}^x}=$

$=e^{x\ln (1+\frac{1}{x})}$

Using Derivative formulas and the multiplication rule and chain rule in Derivative Rules, we get the derivative:

$f'(x)=e^{x\ln (1+\frac{1}{x})}\cdot (1\cdot\ln(1+\frac{1}{x})+x\cdot\frac{1}{1+\frac{1}{x}}\cdot (-\frac{1}{x^2})=$

One can simplify the derivative:

$=e^{x\ln (1+\frac{1}{x})}\cdot ((\ln(1+\frac{1}{x})-\frac{1}{x(1+\frac{1}{x})})=$

By logarithm rules we get:

$={(1+\frac{1}{x})}^x (\ln(1+\frac{1}{x})-\frac{1}{x+1})$

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