Derivative of 5e^x
2026-02-28 10:17 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-5e-to-the-power-x

We can derive the derivative of 5e^x using proofs. To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as:

  1. By First Principle
  2. Using Constant Multiple Rule
  3. Using Exponential Rule

We will now demonstrate that the differentiation of 5e^x results in 5e^x using the above-mentioned methods:

By First Principle

The derivative of 5e^x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of 5e^x using the first principle, we will consider f(x) = 5e^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) = 5e^x, we write f(x + h) = 5e^(x + h).

Substituting these into equation (1), f'(x) = limₕ→₀ [5e^(x + h) - 5e^x] / h = 5 limₕ→₀ [e^x (e^h - 1)] / h

We know that e^h - 1 is approximately h for small h, f'(x) = 5e^x limₕ→₀ h / h = 5e^x

Hence, proved.

Using Constant Multiple Rule

To prove the differentiation of 5e^x using the constant multiple rule, We use the formula: d/dx (cf(x)) = c d/dx f(x) Let c = 5 and f(x) = e^x. d/dx (5e^x) = 5 d/dx (e^x)

Since the derivative of e^x is e^x, = 5e^x

Using Exponential Rule

The exponential rule states that the derivative of e^x is e^x. For the function 5e^x, the derivative is: d/dx (5e^x) = 5 d/dx (e^x) = 5e^x

Thus, the derivative of 5e^x is 5e^x.