Derivatives of Exponentials

Business Calculus · Axiom Academy

LESSON Derivatives of Exponential Functions The amazing properties of e^x and other exponential functions Recall that e 2.71828... is a special irrational number that appears naturally in calculus. What makes it so special? is the only function that equals its own derivative! The function e^x is its own derivative The graph shows f(x) = e^x . At every point, the slope equals the height! When y = e^2 7.39 , the slope is also 7.39 . What about other bases, like 2^x or 10^x ? Natural log of the base appears as a factor When a = e , we get (e) = 1 , so dx [e^x] = e^x 1 = e^x . This is why e is used in calculus! Its derivative has no extra factor, making calculations simpler. When the exponent is a function of x , we need the chain rule: Don't forget to multiply by the derivative of the exponent! Example 1: Find the derivative of f(x) = e^ 3x Example 2: Find the derivative of g(x) = e^ x^2 Exponential functions model growth and decay in business. Their derivatives tell us rates of change . If P dollars are invested at annual rate r (as a decimal), compounded continuously, the value after t years is: The rate of change of the investment: The investment grows at a rate proportional to its current value! Example: 5,000 invested at 6% continuously Click each problem to reveal the solution: Find the derivative of f(x) = e^ 4x - 1 You've mastered exponential derivatives! Next: Derivatives of logarithmic functions!

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