3.9 Derivatives of exponential and logarithmic functions  (Page 4/6)

Key concepts

• On the basis of the assumption that the exponential function $y=b^x,b>0$ is continuous everywhere and differentiable at 0, this function is differentiable everywhere and there is a formula for its derivative.
• We can use a formula to find the derivative of $y=\text{ln}x,$ and the relationship ${\text{log}}_{b}x=\frac{\text{ln}x}{\text{ln}b}$ allows us to extend our differentiation formulas to include logarithms with arbitrary bases.
• Logarithmic differentiation allows us to differentiate functions of the form $y=g{(x)}^{f(x)}$ or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating.

Key equations

• Derivative of the natural exponential function
  $\frac{d}{dx}\left(e^{g(x)}\right)=e^{g(x)}{g}^{\prime }\left(x\right)$
• Derivative of the natural logarithmic function
  $\frac{d}{dx}\left(\text{ln}g\left(x\right)\right)=\frac{1}{g(x)}{g}^{\prime }\left(x\right)$
• Derivative of the general exponential function
  $\frac{d}{dx}\left(b^{g\left(x\right)}\right)=b^{g(x)}{g}^{\prime }\left(x\right)\text{ln}b$
• Derivative of the general logarithmic function
  $\frac{d}{dx}\left({\text{log}}_{b}g\left(x\right)\right)=\frac{g^{\prime }\left(x\right)}{g\left(x\right)\text{ln}b}$

For the following exercises, find $f^{\prime }\left(x\right)$ for each function.