Question

derivative of (x +1)^x + 5^x + 5^5

Answer 1

The only part of this im not sure about is the (x+1)^x. I put (x+1)^x * ln(x+1) on my hw but it was wrong

i thought when you have a variable as an exponent you take the original thing then ln of the base.

Answer 2

\( (x+1)^{x} = e^{x\ln(1+x)}\) <--- use chain rule
\( 5^x = e^{x\ln5}\)

Answer 3

is e^(xln5) the same as 5^x * ln5?

Thats the way my teacher always showed us. And how would that be written my way for the (x+1)^x?

Answer 4

yes
\( e^{\ln 5} = 5\), and \( (e^{\ln 5})^x = 5^x\)

Answer 5

how would i convert \[e ^{xln(1+x)}\] to not have e in it at all. Thats the form i understand better?