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Math Help - Complex Derivative with Natural Log

  1. #1
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    Complex Derivative with Natural Log

    Evaluate D_z[3^z\ln(z+5)]

    I used the product rule in combination with the chain rule.

    Our professor taught us that in order to solve D_z3^z we must rewrite 3^z as e^{ln3^z}. So, using the chain rule, i found that D_z\ln(z+5) \ = \ \ln3 \ \times e^{z\ln3}.

    Continuing on with the product rule...

    3^z \ \times \ D_z[\ln(z+5)] \ + \ \ln(z+5) \ \times \ D_z[3^z].

    Which, rewritten, looks like:

    3^z \ \times \ \frac{1}{z+5} \ + \ \ln(z+5) \ \times \ \ln3 \ \times \ e^{zln3}.

    Now I am not sure if there is a way to rewrite what I have above to make it match this, but the answer we were given is 3^z[\frac{1}{z+5} \ + \ \ln(z+5)\ln3].

    Am I doing something wrong in calculating my derivative, or is there a way to rewrite my answer that I'm not seeing?
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  2. #2
    Member
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    Dec 2009
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    e^{z \ln 3} = e^{\ln(3^z)} = 3^z

    Now, re-write your answer using the above information. You should see a factor common to both terms.
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