Derivatives · Definition

Derivatives of exponentials and logarithms

calculusderivativesexponentials

Before you start you need to know

Exponentials and logarithms have very simple derivatives, especially with base \(e\).

Definition

The function \(e^x\) is its own derivative. For base \(a>0\), \(a^x\) has derivative \(a^x\ln a\), and \(\ln x\) has derivative \(\frac{1}{x}\) for \(x>0\).

\[(e^{x})'=e^{x},\quad(a^{x})'=a^{x}\ln a,\quad(\ln x)'=\frac{1}{x}\]

The exponential \(e^x\) is the function that is its own derivative: its rate of change at each point equals its value. That property defines the number \(e\).

Its inverse, \(\ln x\), has derivative \(1/x\). With them, \((a^x)'=a^x\ln a\) and \((\log_a x)'=\frac{1}{x\ln a}\).

They are the model for growth and decay: populations, compound interest, radioactivity.

Think about it

Differentiate \(e^{3x}\).

\(3e^{3x}\), by the chain rule.

Exercises 10

Differentiate the following functions

  1. \(\displaystyle f(x)=e^{3 x}\)
  2. \(\displaystyle f(x)=2^{x}\)
  3. \(\displaystyle f(x)=\ln{\left(3 x \right)}\)
  4. \(\displaystyle f(x)=5^{x}\)
  5. \(\displaystyle f(x)=e^{-x}\)
  6. \(\displaystyle f(x)=\log_{3}(x^{2}+1)\)
  7. \(\displaystyle f(x)=\frac{e^{x}}{x}\)

Problems

  1. Differentiate \(f(x)=\ln x\) for \(x>0\).
  2. Differentiate \(f(x)=e^{x}\ln x\) for \(x>0\).
  3. Compute \((e^{x})'\) at \(x=0\).

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.