SOLUTION: Evaluate dy/dx when x=-1 given, y=(x+2)^x

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Question 1183844: Evaluate dy/dx when x=-1 given, y=(x+2)^x
Found 2 solutions by Edwin McCravy, robertb:
Answer by Edwin McCravy(20063) About Me  (Show Source):
You can put this solution on YOUR website!
+y=%28x%2B2%29%5Ex

Take natural logs of both sides:

+ln%28y%29=ln%28%28x%2B2%29%5Ex%29

+ln%28y%29=x%2Aln%28x%2B2%29

%22y%27%22%2Fy=x%2A%281%2F%28x%2B2%29%29%2Bln%28x%2B2%29%2A1

%22y%27%22%2Fy=x%2F%28x%2B2%29%2Bln%28x%2B2%29

%22y%27%22=y%2A%28x%2F%28x%2B2%29%2Bln%28x%2B2%29%29

Substitute for y from the original given equation:

%22y%27%22=%28x%2B2%29%5Ex%2A%28x%2F%28x%2B2%29%2Bln%28x%2B2%29%29

%22y%27%28-1%29%22=%28-1%2B2%29%5E%28-1%29%2A%28%28-1%29%2F%28-1%2B2%29%2Bln%28-1%2B2%29%29

%22y%27%28-1%29%22=%281%29%5E%28-1%29%2A%28%28-1%29%2F%281%29%2Bln%281%29%29

%22y%27%28-1%29%22=1%2A%28-1%2B0%29

%22y%27%28-1%29%22=1%2A%28-1%29

%22y%27%28-1%29%22=-1

%22y%27%22 is the same as dy%2Fdx

Edwin

Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
y+=+%28x%2B2%29%5Ex+=+e%5E%28ln%28x%2B2%29%5Ex%29+=+e%5E%28xln%28x%2B2%29%29

===> after applying the chain rule.

===> dy%28-1%29%2Fdx+=%28-1%2B2%29%5E%28-1%29%2A%28ln%28-1%2B2%29+%2B+-1%2F%28-1%2B2%29+%29+=+-1