Definition Derivative Of Exponential Function

Illustration of how the derivative of the exponential function is a multiple of the function where that multiple is the derivative at zero.

Definition derivative of exponential function. An exponential function is a mathematical function in form f x a x where x is a variable and a is a constant which is called the base of the function and it should be greater than 0. This formula is proved on the page definition of the derivative the function y ex is often referred to as simply the exponential function. Besides the trivial case f x 0 the exponential function y ex is the only function whose derivative is equal to itself. The derivative of an exponential function can be derived using the definition of the derivative.

A derivative of an exponential function it is important to note that with the power rule the exponent must be a constant and the base must be a variable while we need exactly the opposite for the derivative of an exponential function. The derivative is the natural logarithm of the base times the original function. Derivatives of exponential functions in order to differentiate the exponential function f x a x f x ax we cannot use power rule as we require the exponent to be a fixed number and the base to be a variable. It means the slope is the same as the function value the y value for all points on the graph.

The graph of the function f x b x where you can enter a value for b is shown by the thick blue curve. Exponential functions the derivative of ln x the derivative of e with a functional exponent the derivative of ln u x. The most commonly used exponential function base is the transcendental number e which is approximately equal to 2 71828. Derivatives of exponential functions involve the natural logarithm function which itself is an important limit in calculus as well as the initial exponential function.

At this point the y value is e 2 7 39. Since the derivative of e x is e x then the slope of the tangent line at x 2 is also e 2 7 39. Let s take the example when x 2. This calculus video tutorial explains how to find the derivative of exponential functions using a simple formula.