Taking a derivative using the chain rule.

Question

I was given the function g(t)=(ln 15)^t

This derivative I know to use the chain rule however I am a little confused on how. Since t is the variable in the exponent would you do ln(ln15)*(ln15)^t or would it be t(ln15)^t-1? And then for the second part of the chain rule how would you take the derivative of the inner? isn't that just a  number so wouldn't that just be 0 making the whole derivative 0?

Answer 1

Let a = ln(15)  [the math will look simpler]

 

Then 

 

g(t) = a^t

 

Take ln() on both sides and get

 

ln(g) = t ln(a) = ln(a) t  [note the right side is of the form kt where k = ln(a) = constant]

 

Take derivative on both sides using logarithmic derivative with chain rule on the left, and derivative of a power on the right, and get

 

g'/g = ln(a)

 

g' = g ln(a)

 

g'(t) = a^t ln(a) = [ln(15)]^t · ln[ln(15)]