Differentiating Exponential & Logarithmic Functions (DP IB Analysis & Approaches (AA): HL): Revision Note

Differentiating exponential & logarithmic functions

What are exponential and logarithmic functions?

• Exponential functions have term(s) where the variable ($x$) is in the power (exponent)

  • In general, these would be of the form $y=a^x$

    • The special case of this is when $a=\mathrm{e}$, i.e.  $y={\mathrm{e}}^x$

• Logarithmic functions have term(s) where logarithms of the variable ($x$) are involved

  • In general, these would be of the form $y={\log }_{a}x$

    • The special case of this is when $a=\mathrm{e}$, i.e.  $y={\log }_{\mathrm{e}}x=\ln x$

What are the derivatives of exponential functions?

• The first two results, of the special cases above, have been met before

  • $f\left(x\right)={\mathrm{e}}^x,f^{\text{'}}\left(x\right)={\mathrm{e}}^x$

  • $f\left(x\right)=\ln x,f^{\text{'}}\left(x\right)=\frac{1}{x}$

  • These are given in the exam formula booklet

• For the general forms of exponentials and logarithms

  • $f\left(x\right)=a^x$

    • $f^{\text{'}}\left(x\right)=a^x\left(\ln a\right)$

  • $f\left(x\right)={\log }_{a}x$

    • $f^{\text{'}}\left(x\right)=\frac{1}{x\ln a}$

  • These are also given in the exam formula booklet

How do I show or prove the derivatives of exponential and logarithmic functions?

• For $y=a^x$

  • Take natural logarithms of both sides, $\ln y=\ln \left(a^x\right)$

  • Use the laws of logarithms, $\ln y=x\ln a$

  • Differentiate implicitly, $\frac{1}{y}\frac{\mathrm{d}y}{\mathrm{d}x}=\ln a$

  • Rearrange, $\frac{\mathrm{d}y}{\mathrm{d}x}=y\ln a$

  • Substitute for $y$, $\frac{\mathrm{d}y}{\mathrm{d}x}=a^x\ln a$

• For $y={\log }_{a}x$

  • Rewrite, $x=a^y$

  • Differentiate with respect to $y$, using the above result, $\frac{\mathrm{d}x}{\mathrm{d}y}=a^y\ln a$

  • Use $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}y}}}$, $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{a^y\ln a}$

  • Substitute for $y$, $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{a^{{\log }_{a}x}\ln a}$

  • Simplify, $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{x\ln a}$

What do the derivatives of exponentials and logarithms look like with linear functions of x?

• For linear functions of the form $px+q$

  • $f\left(x\right)=a^{px+q}$

    • $f^{\text{'}}\left(x\right)=p{a}^{px+q}\left(\ln a\right)$

  • $f\left(x\right)={\log }_a(px+q)$

    • $f^{\text{'}}\left(x\right)=\frac{p}{\left(px+q\right)\ln a}$

  • These are not in the formula booklet

    • they can be derived from chain rule

    • they are not essential to remember