Derivative of 7^x
2026-02-28 23:52 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-7%5Ex

We can derive the derivative of 7x using proofs.

To show this, we will use the properties of logarithms and exponentials along with the rules of differentiation. There are several methods we use to prove this, such as:

By Definition of Derivative

Using Logarithmic Differentiation

By Definition of Derivative

The derivative of 7x can be proved using the definition of the derivative, which expresses the derivative as the limit of the difference quotient. To find the derivative of 7^x using the definition, we will consider f(x) = 7x.

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

Given that f(x) = 7x, we write f(x + h) = 7(x + h).

Substituting these into equation (1), f'(x) = lim_(h→0) [7(x + h) - 7^x] / h = lim_(h→0) [7x * 7h - 7x] / h = 7x * lim_(h→0) [7h - 1] / h

We recognize this limit as the definition of the derivative of an exponential function, which results in: f'(x) = 7x ln(7)

Hence, proved.

Using Logarithmic Differentiation

To prove the differentiation of 7x using logarithmic differentiation,

Let y = 7x

Take the natural logarithm on both sides: ln(y) = ln(7x)

Using logarithm properties, ln(y) = x ln(7)

Differentiate both sides with respect to x: (1/y) dy/dx = ln(7) dy/dx = y ln(7)

Substitute y = 7x: dy/dx = 7x ln(7)

Thus, we have shown that the derivative of 7x is (7x)ln(7).