Derivative of 3^2x
2026-02-28 06:04 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-3^2x

We can derive the derivative of 3^2x using various proofs. To show this, we will use the properties of exponential functions along with the rules of differentiation. There are several methods 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 3^2x results in 2 * 3^2x * ln(3) using the above-mentioned methods:

By First Principle

The derivative of 3^2x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of 3^2x using the first principle, consider f(x) = 3^2x. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h

Given that f(x) = 3^2x, we write f(x + h) = 3^2(x + h).

Substituting these into the equation, f'(x) = limₕ→₀ [3^2(x + h) - 3^2x] / h = limₕ→₀ [3^2x * 3^2h - 3^2x] / h = limₕ→₀ [3^2x * (3^2h - 1)] / h = 3^2x * limₕ→₀ [(3^2h - 1) / h]

Using the properties of limits and the fact that limₕ→₀ (e^h - 1)/h = ln(e), f'(x) = 3^2x * ln(3) * 2 = 2 * 3^2x * ln(3).

Hence, proved.

Using Chain Rule

To prove the differentiation of 3^2x using the chain rule, We use the formula: If y = a^u, then dy/dx = a^u * ln(a) * du/dx Let u = 2x, then y = 3^u = 3^2x

Using the chain rule, dy/dx = 3^2x * ln(3) * d(2x)/dx = 2 * 3^2x * ln(3).

We will now prove the derivative of 3^2x using the exponential rule. The step-by-step process is demonstrated below: Here, we use the formula, If y = a^x, then dy/dx = a^x * ln(a)

But since we have 3^2x instead of 3^x, y = 3^2x

Let u = 2x, then dy/dx = 3^u * ln(3) * du/dx = 2 * 3^2x * ln(3).