Finding the Derivative of e^x | A Fundamental Result in Calculus

Derivative of e^x

The derivative of e^x, denoted as d/dx (e^x) or e^x’, is a fundamental result in calculus

The derivative of e^x, denoted as d/dx (e^x) or e^x’, is a fundamental result in calculus. To find the derivative of e^x, we can use the exponential function’s special property.

The exponential function e^x is defined as the sum of an infinite series:

e^x = 1 + x/1! + (x^2)/2! + (x^3)/3! + …

Each term in the series is related to the previous term by dividing by the next positive integer.

Now, let’s find the derivative of e^x using this definition:

For the first term, the derivative of 1 is zero.

For the second term, the derivative of x/1! is 1 because the derivative of any polynomial function is simply the coefficient of the term multiplied by its exponent.

For the third term, the derivative of (x^2)/2! is x/1! = x.

Similarly, for the fourth term, the derivative of (x^3)/3! is (x^2)/2! = x^2/2.

This pattern continues for all the terms in the series. We observe that each term’s derivative is simply the term itself, but with the exponent reduced by one:

d/dx (e^x) = d/dx (1) + d/dx (x/1!) + d/dx ((x^2)/2!) + d/dx ((x^3)/3!) + …
= 0 + 1 + x + (x^2)/2 + (x^3)/3! + …

Simplifying further, we can represent the derivative of e^x as:

d/dx (e^x) = 1 + x + (x^2)/2 + (x^3)/3! + …

So, the derivative of e^x is just e^x itself, meaning that the rate at which e^x changes is equal to e^x itself.

In summary: The derivative of e^x is e^x.

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