How to Find the Derivative of f(x) = e^x Using Differentiation Rules

Derivative of e^x

To find the derivative of the function f(x) = e^x, we can use the rules of differentiation

To find the derivative of the function f(x) = e^x, we can use the rules of differentiation. In this case, we will differentiate with respect to x.

The derivative of any constant raised to the power of x can be found using the chain rule. The chain rule states that if we have a composition of functions, the derivative is found by multiplying the derivative of the outer function by the derivative of the inner function.

The function e^x can be written as the composition of the exponential function f(u) = e^u and the identity function g(x) = x. Hence, f(g(x)) = e^(g(x)) = e^x.

To find the derivative, we need to differentiate e^x with respect to x. Since the inner function g(x) = x is simply x in this case, the derivative of g(x) is 1. Now, let’s differentiate the outer function using the chain rule.

Let u = g(x) = x, and f(u) = e^u. Applying the chain rule, the derivative of f(u) with respect to u is f'(u) = e^u. Now, we can multiply this by the derivative of the inner function g(x) to get the final result.

Therefore, the derivative of f(x) = e^x with respect to x is:

f'(x) = f'(u) * g'(x)
= (e^u) * 1
= e^x

So, the derivative of e^x with respect to x is e^x.

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