Question

Logarithmic differantiation,
http://screencast.com/t/fwBW2D9z9E

Can you help me out on this please

Answer 1

when there is an \(x\) in the exponent and want to compute a limit or a derivative, always resort to this technique:
\[
a^{h(x)} = e^{\ln a^{h(x)}} = e^{h(x) \ln a}
\]
from here you should be able to do something.

Answer 2

were you able to do it??

Answer 3

do you know the "more simple" formula: \((a^x)' = a^x \ln a\) ? (\(a>0\))
knowing this you don't need the above trick. (you have to know the derivation of a compound function formula)

Answer 4

Let y=f(x)
\[\Large y=\pi^{sinx}\]
ln both sides
\[\ln(y)=\sin(x)\ln(\pi)\]
Differentiate each term
\[\frac{1}{y}y'=\ln(\pi)\cos(x) \\ \\ y'=yln(\pi)\cos(x) \\ \\ y'=\pi^{sinx}\ln(\pi)\cos(x)\]
Got it?

Answer 5

If that's it yea easy enough

Answer 6

("always" means that it works, but there can be simpler ways depending on the problem)

\[
(\pi^{\sin x})' = (\pi^{\sin x} \ln \pi) (\sin x)' = (\pi^{\sin x} \ln \pi) (\cos x)
\] (derivation by parts.)