Calc Midterm, First Year University (derivative related stuff)

[y2klimen]

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

...?

 

[snipez90]

If f(x) = x^x, then f'(x) =/= x*x^(x-1). That rule works if your exponent is a number, not a function. Rewrite x^x in exponential form and then differentiate using the chain rule to find f'(x).

 

[HallsofIvy]

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

...?

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).

If y= x^x then ln(y)= x ln(x). Use the chain rule to find dy/dx.