The Derivative of aˣ and the Natural Logarithm

Two of the most important results in differential calculus are

d/dx(ln x) = 1/x
d/dx(aˣ) = aˣ ln a.

These formulas are closely connected. One describes the derivative of the natural logarithm, while the other gives the derivative of an exponential function with positive base. Together, they reveal the deep relationship between logarithms, exponentials, inverse functions, and differentiation.

The Derivative of aˣ and the Natural Logarithm

1. Derivative of the natural logarithm

Begin with

ln x = y.

This is equivalent to

eʸ = x.

Differentiate both sides with respect to y:

dx/dy = eʸ.

Now invert this result:

dy/dx = 1/eʸ.

Since eʸ = x, substitute back:

dy/dx = 1/x.

Therefore,

d/dx(ln x) = 1/x, for x > 0.

So the derivative of the natural logarithm is the reciprocal of x.

2. Derivative of the exponential function aˣ

Now let

y = aˣ,

where a > 0 and a ≠ 1.

Take logarithms to base a:

logₐ y = x.

Now apply the change-of-base formula:

ln y / ln a = x.

Differentiate both sides with respect to y:

dx/dy = (1/ln a)(1/y).

So

dx/dy = 1/(y ln a).

Inverting gives

dy/dx = y ln a.

Since y = aˣ, substitute back:

dy/dx = aˣ ln a.

Therefore,

d/dx(aˣ) = aˣ ln a, for a > 0 and a ≠ 1.

This is the general derivative formula for an exponential function with positive base.

3. The special case a = e

When a = e, the formula becomes especially elegant:

d/dx(eˣ) = eˣ ln e = eˣ,

because ln e = 1.

This means that eˣ is the unique exponential function whose derivative is exactly equal to itself.

4. Why the factor ln a appears

Exponential functions with different bases grow at different rates. The derivative of aˣ is always proportional to aˣ itself, but the constant of proportionality depends on the base. That constant is ln a.

This is why e is so important in calculus. Since ln e = 1, the exponential function eˣ differentiates in the simplest possible way.

5. Final results

The derivation establishes the following two fundamental formulas:

d/dx(ln x) = 1/x, for x > 0
d/dx(aˣ) = aˣ ln a, for a > 0 and a ≠ 1.

These results are central to calculus. The natural logarithm differentiates to 1/x, while the exponential function aˣ differentiates to itself multiplied by ln a. Together, they show how logarithmic and exponential functions are tied together through inverse structure and differentiation.

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