Calculus
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Boundless Calculus
Inverse Functions and Advanced Integration
Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions
Calculus Textbooks Boundless Calculus Inverse Functions and Advanced Integration Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions
Calculus Textbooks Boundless Calculus Inverse Functions and Advanced Integration
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus
Concept Version 8
Created by Boundless

Derivatives of Exponential Functions

The derivative of the exponential function is equal to the value of the function.

Learning Objective

  • Solve for the derivatives of exponential functions


Key Points

    • $e^x$ is its own derivative: $\frac{d}{dx}e^{x} = e^{x}$.
    • If a variable's growth or decay rate is proportional to its size, then the variable can be written as a constant times an exponential function of time.
    • For any differentiable function $f(x)$, $\frac{d}{dx}e^{f(x)} = f'(x)e^{f(x)}$.

Terms

  • tangent

    a straight line touching a curve at a single point without crossing it there

  • e

    the base of the natural logarithm, $2.718281828459045\dots$

  • exponential

    any function that has an exponent as an independent variable


Full Text

The importance of the exponential function in mathematics and the sciences stems mainly from properties of its derivative. In particular:

$\dfrac{d}{dx}e^{x} = e^{x}$

That is to say, $e^x$ is its own derivative. 

Graph of an Exponential Function

Graph of the exponential function illustrating that its derivative is equal to the value of the function. From any point $P$ on the curve (blue), let a tangent line (red), and a vertical line (green) with height $h$ be drawn, forming a right triangle with a base $b$ on the $x$-axis. Since the slope of the red tangent line (the derivative) at $P$ is equal to the ratio of the triangle's height to the triangle's base (rise over run), and the derivative is equal to the value of the function, $h$ must be equal to the ratio of $h$ to $b$. Therefore, the base $b$ must always be $1$.

Functions of the form $ce^x$ for constant $c$ are the only functions with this property. 

Other ways of saying this same thing include: 

  • The slope of the graph at any point is the height of the function at that point.
  • The rate of increase of the function at $x$ is equal to the value of the function at $x$.
  • The function solves the differential equation $y' = y $.
  • $e^x$ is a fixed point of derivative as a functional.

If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth, continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time. Explicitly for any real constant $k$, a function $f: R→R$ satisfies $f′ = kf $ if and only if $f(x) = ce^{kx}$ for some constant $c$.

Furthermore, for any differentiable function $f(x)$, we find, by the chain rule:

$\displaystyle{\frac{d}{dx}e^{f(x)} = f'(x)e^{f(x)}}$

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