Question

how do you take the derivative of a number to a power to a power?

Answer 1

first get it to "logarithms" soo that the exponent comes down and multiplies. Have any particular example?

Answer 2

yes. hold on

Answer 3

\[7-(1.01)^{-t^{2}}\]

Answer 4

ok.
\[y=7-(1.01)^{-t^2}\\
7-y=(1.01)^{-t^2}\\
\ln(7-y)=-t^2\ln(1.01)\\
{\rm differentiate}\\
{1\over 7-y}(-y')=-2t\ln(1.01)\\
\boxed{y'=(-2t)(y-7)\ln(1.01)\\
y'=2t(1.01)^{-t^2}\ln(1.01)
}\]

Answer 5

do you follow?

Answer 6

i do til the third line

Answer 7

then I differentiated. Notice that on the left side, I take the derivative of the log and then have to apply the chain rule

Answer 8

i dont understand how you got to line 3

Answer 9

\[
7-y=(1.01)^{-t^2}\\
\text{take log on both sides}\\
\ln(7-y)=\ln\left(1.01^{-t^2}\right)\\
\text{use the property: }\quad \ln(a^m)=m\times\ln(a)\\
\ln(7-y)=-t^2\ln(1.01)
\]

Answer 10

oh. ok. i understand now

Answer 11

thank you