Exponential functions model quantities that grow or decay at rates proportional to their current size. Their derivatives are central in calculus because they connect a function's value directly to its instantaneous rate of change. The most important exponential function is y = e^x, since its slope at every point equals its height.
This special property makes e the natural base for growth, decay, finance, biology, physics, and differential equations.
For a general exponential function y = a^x, the derivative includes a scaling factor ln(a), so d/dx(a^x) = a^x ln(a). When the exponent is itself a function, the chain rule multiplies by the derivative of the exponent, giving d/dx(e^{u}) = e^{u}u'. These formulas explain why tangent lines on the graph of e^x get steeper as x increases.
They also show that exponential rates are not constant, but change in proportion to the current value.
Key Facts
- d/dx(e^x) = e^x
- d/dx(a^x) = a^x ln(a), for a > 0 and a != 1
- d/dx(e^{u(x)}) = e^{u(x)}u'(x)
- d/dx(a^{u(x)}) = a^{u(x)} ln(a) u'(x)
- The slope of y = e^x at x = c is e^c.
- For y = Ce^{kx}, the derivative is dy/dx = kCe^{kx} = ky.
Vocabulary
- Exponential function
- A function in which the variable appears in the exponent, such as f(x) = a^x with a positive base a.
- Natural exponential function
- The function f(x) = e^x, whose derivative is exactly itself.
- Derivative
- The instantaneous rate of change of a function, represented by the slope of its tangent line.
- Natural logarithm
- The logarithm with base e, written ln(x), which appears in the derivative of a^x.
- Chain rule
- A rule for differentiating composite functions by multiplying the derivative of the outside function by the derivative of the inside function.
Common Mistakes to Avoid
- Writing d/dx(a^x) = a^x is wrong because only the base e has this exact derivative.
- Forgetting the factor ln(a) in d/dx(a^x) is wrong because changing the base changes the growth rate scale.
- Dropping the chain rule factor in d/dx(e^{u(x)}) is wrong because the exponent may be changing faster or slower than x.
- Treating exponential functions like power functions is wrong because d/dx(x^n) = nx^{n-1} does not apply when x is in the exponent.
Practice Questions
- 1 Find the derivative of f(x) = 7e^x at x = 2. Give your answer exactly.
- 2 Find dy/dx for y = 3^{2x - 5}.
- 3 Explain why the tangent slope of y = e^x increases as x increases, and connect your explanation to the derivative formula.