How to differentiate an Exponential function with irrational exponent using the chain rule & 2e^(√x)
To differentiate an exponential function with base e, you can use the chain rule. The derivative of the exponential function e^x with respect to x is itself, and the chain rule accounts for the fact that the exponent (or index) is a function of x. The general form is:
d/dx(e^u) = e^u × du/dx
For the case where u = x, the derivative of e^x is simply e^x.
So, if you have a function f(x) = e^g(x), where g(x) is another function of x, the derivative f'(x) is given by:
f'(x) = e^g(x) × g'(x)
Here's a simple example to illustrate:
Suppose you have the function f(x) = e^(2x). Let u = 2x, then g'(x) = 2. Applying the chain rule:
f'(x) = e^(2x) × 2
So, the derivative of e^(2x) with respect to x is 2e^(2x).
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