Utilizing Logarithmic Differentiation to Find the Derivative of a^x

d/dx a^x

To find the derivative of a function of the form a^x, where a is a constant, we can use the natural logarithm

To find the derivative of a function of the form a^x, where a is a constant, we can use the natural logarithm.

The derivative of a function f(x) = a^x with respect to x can be found using logarithmic differentiation. The idea is to take the natural logarithm (ln) of both sides of the equation, apply properties of logarithms to simplify, differentiate implicitly, and then solve for the derivative.

Let’s go through the steps:

Step 1: Take the natural logarithm (ln) of both sides of the equation:
ln(f(x)) = ln(a^x)

Step 2: Use the logarithmic property ln(ab) = b * ln(a) to simplify the equation:
ln(f(x)) = x * ln(a)

Step 3: Differentiate implicitly with respect to x:
(d/dx) ln(f(x)) = (d/dx) (x * ln(a))

To differentiate the left side, we can use the chain rule:
(d/dx) ln(f(x)) = (d/dx) f(x) / f(x)

Since f(x) = a^x, we have:
(d/dx) ln(f(x)) = (d/dx) (a^x) / a^x

Step 4: Differentiate the right side:
Using the chain rule, we find:
(d/dx) (a^x) = (ln(a)) * a^x

Step 5: Substitute the results back into the equation:
(d/dx) f(x) / f(x) = (ln(a)) * a^x / a^x

Step 6: Simplify:
(d/dx) f(x) / f(x) = ln(a)

Step 7: Solve for (d/dx) f(x):
(d/dx) f(x) = f(x) * ln(a)

Since f(x) = a^x, we substitute back:
(d/dx) a^x = a^x * ln(a)

Therefore, the derivative of a^x with respect to x is a^x times the natural logarithm of a, which can be written as:
(d/dx) a^x = a^x * ln(a)

This is the final answer.

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