Differentiating Exponential Functions with Other Bases

Key Questions

  • How do I find f'(x) for f(x)=5^x ?

    Answer:

    f'(x)=ln5*5^x

    Explanation:

    Let y=f(x)=5^x

    then lny=xln5 and differentiating we get

    1/y(dy)/(dx)=ln5

    or (dy)/(dx)=ln5*y=ln5*5^x

    Shwetank Mauria · 3 · Apr 4 2018
  • How do I find f'(x) for f(x)=3^-x ?

    The answer is f'(x)=-3^(-x)ln3.

    First, step is a change of base:

    f(x)=3^(-x)
    =e^(ln 3^(-x))
    =e^(-xln3)

    With the proper base e, we can just use the chain rule:

    f'(x)=e^(-xln3)(-ln3)
    =3^(-x)(-ln3)

    rearrange and you will get the same answer as the first line.

    The other option is to use the general exponential differentiation rule (if you can remember it):

    f(x)=a^u
    f'(x)=a^u ln a (du)/(dx)

    Amory W. · · Sep 7 2014
  • What is the derivative of f(x)=b^x ?

    This is the exponential function of base b (where b>0 should be assumed). It can be thought of as b^x=e^(xln(b)), so that, using the Chain Rule (See Chain Rule ) and the fact that (e^x)'=e^x (see Exponentials with Base e ) yields (b^x)'=e^(xln(b))\times ln(b)=b^x\times ln(b)
    (see Exponential functions ).

    Calculus V. · · Jul 23 2014

Questions

Differentiating Exponential Functions