The discussion centers on finding the constant 'a' such that the derivative of the function f(x) = x^x equals twice the function value at that point, specifically f'(a) = 2f(a). Participants emphasize the need to differentiate x^x correctly using the chain rule, noting that the derivative f'(x) is not simply x*x^(x-1) but requires rewriting the function in exponential form. The correct differentiation leads to the conclusion that ln(y) = x ln(x) must be applied to derive dy/dx accurately.
PREREQUISITESFirst-year university students studying calculus, particularly those tackling derivatives of exponential functions, as well as educators looking for examples of common misunderstandings in differentiation.
No that is not right. You cannot treat a (x) as a variable in one case (the base) and a constant in the other (the exponent).y2klimen said:Homework Statement
If f(x)=x^x for x>0, find the constant a such that f'(a)=2f(a)
Homework Equations
The Attempt at a Solution
f'(a)=a*a^(a-1)
=a^a
2f(a)=2a^a
2a^a=a^a
ln2a^a=lna^a
aln2a=alna
...?