SOLUTION: find the derivative of the function {{{y = 2 ^ (sin(pi*x))}}}

Algebra ->  Exponents -> SOLUTION: find the derivative of the function {{{y = 2 ^ (sin(pi*x))}}}      Log On


   



Question 353582: find the derivative of the function
y+=+2+%5E+%28sin%28pi%2Ax%29%29

Found 3 solutions by vleith, Fombitz, Edwin McCravy:
Answer by vleith(2983) About Me  (Show Source):
You can put this solution on YOUR website!
Use this URL. Then click Show Steps to see the work
http://www.wolframalpha.com/input/?i=+y%3D2+^+%28sin+pi+x%29++derivative

Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Use the chain rule,
Let u=sin%28pi%2Ax%29
dy%2Fdx=%28dy%2Fdu%29%2A%28du%2Fdx%29
dy%2Fdu=ln%282%29%2A2%5Eu
du%2Fdx=cos%28pi%2Ax%29%2Api
Put it all together,
dy%2Fdx=ln%282%29%2A2%5E%28u%29%2Api%2Acos%28pi%2Ax%29
dy%2Fdx=ln%282%29%2Api%2A2%5E%28sin%28pi%2Ax%29%29%2Acos%28pi%2Ax%29

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
y+=+2+%5E+%28sin%28pi%2Ax%29%29

Take natural logs of both sides
ln%28y%29+=+ln%282+%5E+%28sin%28pi%2Ax%29%29%29
Use a rule of logarithms of exponentials on the right side:
ln%28y%29+=+sin%28pi%2Ax%29ln%282%29
ln%28y%29+=+ln%282%29sin%28pi%2Ax%29
[I wrote it that way so your would recognize that ln(2) is just a constant.]
Find the derivative of both sides, remembering that ln(2) is just a
constant and that we must use the chain rule on the right side by
taking the derivitive of the "inside":
%28dy%2Fdx%29%2Fy+=+ln%282%29cos%28pi%2Ax%29%2Api
%28dy%2Fdx%29%2Fy+=+pi%2Aln%282%29cos%28pi%2Ax%29
Multiply both sides by y
dy%2Fdx+=+y%2Api%2Aln%282%29cos%28pi%2Ax%29
To get the derivative in terms of x only, we go back to
the original equation y+=+2+%5E+%28sin%28pi%2Ax%29%29 and
substitute for y
dy%2Fdx+=+%282+%5E+%28sin%28pi%2Ax%29%29%29%2Api%2Aln%282%29cos%28pi%2Ax%29
Writing the constants first, and the more complicated factor last:
dy%2Fdx+=+pi%2Aln%282%29cos%28pi%2Ax%292+%5E+%28sin%28pi%2Ax%29%29