The Derivative of f(x) = a^x using Logarithmic Differentiation: Simplified Step-by-Step Guide

Derivative of a^x

To find the derivative of the function f(x) = a^x, we can use the concept of logarithmic differentiation

To find the derivative of the function f(x) = a^x, we can use the concept of logarithmic differentiation. First, take the natural logarithm of both sides of the equation:

ln(f(x)) = ln(a^x)

Using the properties of logarithms, we can move the exponent down in front:

ln(f(x)) = x * ln(a)

Now, differentiate both sides of the equation with respect to x using the chain rule:

(d/dx) ln(f(x)) = (d/dx) (x * ln(a))

The derivative of ln(f(x)) can be found using the chain rule:

(d/dx) ln(f(x)) = (d/dx) f(x) / f(x)

The right-hand side of the equation can be simplified as follows:

(d/dx) f(x) = (d/dx) (a^x)

Next, let’s find the derivative of (a^x):

Using the chain rule, the derivative of a^x with respect to x is given by:

(d/dx) (a^x) = (a^x) * ln(a)

Now, substituting this into our derivative equation:

(d/dx) ln(f(x)) = (d/dx) f(x) / f(x)

(a^x) * ln(a) = (d/dx) f(x) / f(x)

To find the derivative (d/dx) f(x), we can multiply both sides of the equation by f(x):

(a^x) * ln(a) * f(x) = (d/dx) f(x)

At this point, the derivative is written in terms of f(x). To express it solely in terms of x, we can substitute f(x) with a^x:

(d/dx) f(x) = (a^x) * ln(a) * a^x

Simplifying further:

(d/dx) f(x) = a^x * ln(a) * a^x

Thus, the derivative of f(x) = a^x is:

f'(x) = a^x * ln(a) * a^x

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