Derivative of 6^x
2026-02-28 23:51 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-6%5Ex

We can derive the derivative of 6^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 Logarithmic Differentiation

We will now demonstrate that the differentiation of 6^x results in 6^x ln(6) using the above-mentioned methods:

By First Principle

The derivative of \(6^x\) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of\( 6^x\) using the first principle, we will consider f(x) =\( 6^x\).

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

Given that f(x) = \(6^x\), we write f(x + h) = \(6^(x + h)\).

Substituting these into equation (1), f'(x) = limₕ→₀ [\(6^(x + h) - 6^x\)] / h = limₕ→₀ [\(6^x (6^h - 1)\)] / h = \(6^x\) limₕ→₀ [\((6^h - 1) / h\)]

Using the limit property, limₕ→₀ [\((6^h - 1) / h\)] = ln(6). f'(x) = \(6^x\) ln(6)

Hence, proved.

Using Chain Rule

To prove the differentiation of 6^x using the chain rule, We express 6^x as e^(x ln(6)).

Let u = x ln(6). Then, 6^x = e^u. By the chain rule: d/dx [e^u] = e^u (du/dx)

So we get, d/dx (6^x) = e^(x ln(6)) * ln(6)

Substituting back the expression for e^(x ln(6)) gives us: d/dx (6^x) = 6^x ln(6).

Using Logarithmic Differentiation

We will now prove the derivative of \(6^x \)using logarithmic differentiation.

The step-by-step process is demonstrated below: Consider y = \(6^x\)

Taking the natural logarithm of both sides gives ln(y) = ln(\(6^x\)) = x ln(6)

Differentiating both sides with respect to x gives: (1/y) dy/dx = ln(6)

Thus, dy/dx = y ln(6) = \(6^x \)ln(6)

Therefore, the derivative of the function y =\( 6^x\) is \(6^x\) ln(6).