Derivative of 8^x
2026-02-28 23:53 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-8%5Ex

We can derive the derivative of 8^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:

  1. By First Principle
  2. Using Chain Rule
  3. Using Exponential Rule

We will now demonstrate that the differentiation of 8^x results in 8^x ln(8) using the above-mentioned methods:

By First Principle The derivative of 8^x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of 8^x using the first principle, we will consider f(x) = 8^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) = 8^x, we write f(x + h) = 8^(x + h).

Substituting these into equation (1), f'(x) = limₕ→₀ [8^(x + h) - 8^x] / h = limₕ→₀ [8^x * 8^h - 8^x] / h = 8^x * limₕ→₀ [8^h - 1] / h

Using the limit property, we have limₕ→₀ [8^h - 1] / h = ln(8) Thus, f'(x) = 8^x * ln(8)

Hence, proved.

Using Chain Rule To prove the differentiation of 8^x using the chain rule, Consider y = 8^x = e^(x ln(8)) Let u = x ln(8), then y = e^u

Applying the chain rule: dy/dx = dy/du * du/dx dy/du = e^u = e^(x ln(8)) = 8^x du/dx = ln(8)

Therefore, dy/dx = 8^x * ln(8)

Using Exponential Rule We use the standard rule for differentiating exponential functions:

If y = a^x, then dy/dx = a^x ln(a) Applying this to y = 8^x, we get: dy/dx = 8^x ln(8)

This shows the derivative of 8^x is 8^x ln(8).