Derivative of Exponential Functions
f'(x) = e^x
The derivative of an exponential function is a straightforward process when you understand the basic rules of differential calculus. These rules tell us how to differentiate any function, including an exponential function.
In general, the derivative of an exponential function is proportional to the function itself. Specifically, if we have a function f(x) = e^x, then its derivative f'(x) is given by:
f'(x) = e^x
This means that the slope of the tangent to the curve y = e^x at any point x is just the value of e^x at that point. In other words, as x increases, the value of e^x increases, and so does the slope of the tangent to the curve y = e^x.
The same rule applies to exponential functions with a base other than e. For example, if we have a function f(x) = a^x, where a is a constant greater than 0, then its derivative f'(x) is given by:
f'(x) = a^x * ln(a)
This formula may look a bit intimidating, but it simply tells us that the derivative of a function with base a is proportional to the function itself, but multiplied by the natural logarithm of the base a.
So, when differentiating exponential functions, the basic steps are:
1. Identify the base of the exponential function (this is usually e or some other constant).
2. Apply the rule for derivatives of exponential functions with that base.
3. Simplify the expression if possible.
These rules are useful for solving many types of problems involving derivatives of exponential functions.
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