Exercise
Find the derivative of the following function:
f(x)={(1+\frac{1}{x})}^x
Final Answer
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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