Derivative of Exponents
2026-02-28 11:25 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-exponents

We can derive the derivative of \(e^x\) using proofs.

To show this, we will use the properties of exponential functions along with the rules of differentiation.

There are several methods we use to prove this, such as:

By First Principle

Using Chain Rule

Using Product Rule

We will now demonstrate that the differentiation of \(e^x\) results in e^x using the above-mentioned methods:

By First Principle

The derivative of \(e^x\) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of \(e^x\) using the first principle, we will consider f(x) = \(e^x\).

Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)

Given that f(x) = \(e^x\), we write f(x + h) = \(e^(x + h)\).

Substituting these into equation (1), f'(x) = limₕ→₀ [\(e^(x + h) - e^x\)] / h = limₕ→₀ [\(e^x · (e^h - 1)\)] / h = \(e^x \)· limₕ→₀ [\((e^h - 1)\) / h]

Using the limit property, limₕ→₀ (\(e^h - 1\)) / h = 1. f'(x) = \(e^x\) · 1 = \(e^x\)

Hence, proved.

Using Chain Rule

To prove the differentiation of \(e^x \)using the chain rule,

We use the formula: If y = \(e^(u)\) where u = x, then dy/dx = dy/du · du/dx = \(e^u\) · 1 = \(e^x\)

Using Product Rule

We will now prove the derivative of \(e^x\) using the product rule.

The step-by-step process is demonstrated below:

Here, we consider the function as a product of two identical exponential functions: \(e^x · e^0 = e^x · 1\)

Using the product rule formula: d/dx [u·v] = u'·v + u·v' u = \(e^x\) and v = \(e^0\) u' = d/dx (\(e^x\)) = \(e^x \)v' = d/dx (\(e^0\)) = 0 d/dx (\(e^x\)) = \(e^x · 1 + e^x · 0 = e^x\)

Thus, d/dx (\(e^x\)) =\( e^x\).