Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 06

MATH100 A December 2023

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Question 06
Let ${\textstyle f(x)=x^{x}}$. Find ${\textstyle f'(e)}$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Do you know of a function that you could apply to ${\textstyle f(x)=x^{x}}$ that would simplify the expression so there is no longer a variable in the exponent?

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. None of the usual derivative rules we know will help with this function, so we will need to transform it into a form that we can use derivative rules with! The issue is the variable in the exponent and the base, so if we apply the logarithm to both sides of the expression, we can bring the variable in the exponent down: $${\begin{aligned}\ln(f(x))&=\ln(x^{x})\\\ln(f(x))&=x\ln(x).\end{aligned}}$$ Note: we need to make sure ${\textstyle f(x)\neq 0}$, otherwise ${\textstyle \ln(f(x))}$ will be undefined. Luckily, we are interested in the domain where ${\textstyle x>0}$, and ${\textstyle f(x)}$ does not equal zero over this domain. Now, use implicit differentiation: $${\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}\ln(f(x))&={\frac {\mathrm {d} }{\mathrm {d} x}}x\ln(x)\\{\frac {1}{f(x)}}f'(x)&=\ln(x)+x({\frac {1}{x}})\\f'(x)&=f(x)(\ln(x)+1)\\f'(x)&=x^{x}(\ln(x)+1)\end{aligned}}$$ Sub in ${\textstyle x=e}$: ${\textstyle f'(e)=e^{e}(\ln(e)+1)=2e^{e}}$.