Derivative of e^x: Understanding the Fundamental Result and Calculation Methods in Calculus

Derivative of e^x

The derivative of e^x is a fundamental result in calculus

The derivative of e^x is a fundamental result in calculus. To find the derivative of e^x, we can use the definition of the derivative or use basic differentiation rules.

1. Using the definition of the derivative:

The derivative of a function f(x) is defined as the limit of the difference quotient as h approaches zero:

f'(x) = lim(h->0) [f(x+h) – f(x)] / h

Let’s apply this definition to e^x:

f(x) = e^x

f(x+h) = e^(x+h)

Substituting these values into the difference quotient:

f'(x) = lim(h->0) [e^(x+h) – e^x] / h

Using algebraic manipulation, we can rewrite this expression as:

f'(x) = lim(h->0) e^x [e^h – 1] / h

As h approaches zero, the expression e^h – 1 also approaches zero. Therefore, taking the limit as h approaches zero, we get:

f'(x) = e^x * 1

Finally, the derivative of e^x is e^x.

2. Using basic differentiation rules:

Another way to find the derivative of e^x is by using basic differentiation rules. The exponential function e^x has a special property: its derivative is equal to the function itself.

In other words, d/dx(e^x) = e^x

This property is true for all values of x, and it can be proven using calculus. Therefore, we can simply state that the derivative of e^x is e^x without needing to go through the above steps.

So, in summary, the derivative of e^x is e^x.

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