SOLUTION: Derivative of y = 2^πx

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Question 1140601: Derivative of y = 2^πx
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
Derivative of y+=+2%5E%28pi%2Ax%29+

%28d%2Fdx%29+%282%5E%28pi%2Ax%29+%29

Apply exponent rule : a%5Eb=e%5E%28b%2Aln+%28a%29%29:

2%5E%28pi%2Ax%29+=e%5E%28b%2Aln+%28a%29%29

=%28d%2Fdx%29+%28e%5E%28b%2Aln+%28a%29%29%29

Apply the chain rule : df%28u%29%2Fdx=+%28df%2Fdu%29%2A%28du%2Fdx%29
where f=e%5Eu, u=pi%2Ax%2Aln%282%29

=%28d%2Fdu%29%28e%5Eu%29%2A%28d%2Fdx%29%28pi%2Ax%2Aln%282%29%29
=e%5Eu%2Api%2Aln%282%29

Substitute back :u=+pi+%2Ax%2Aln+%282%29:

=e%5E%28pi+%2Ax%2Aln+%282%29%29%2Api%2Aln%282%29

since e%5E%28pi+%2Ax%2Aln+%282%29%29=+2%5E%28pi%2Ax%29

=2%5E%28pi%2Ax%29+%2Api%2Aln%282%29

so, %28d%2Fdx%29+%282%5E%28pi%2Ax%29+%29=2%5E%28pi%2Ax%29+%2Api%2Aln%282%29