How to Find the Derivative of f(x) = a^x Using Logarithmic Differentiation

Dx {a^x}=?

To find the derivative of the function f(x) = a^x with respect to x, we can use the logarithmic differentiation method

To find the derivative of the function f(x) = a^x with respect to x, we can use the logarithmic differentiation method.

Step 1: Start with the equation f(x) = a^x.

Step 2: Take the natural logarithm (ln) of both sides of the equation: ln(f(x)) = ln(a^x).

Step 3: Apply the logarithm rules to simplify the equation:
ln(f(x)) = x * ln(a).

Step 4: Differentiate both sides of the equation with respect to x:
(d/dx) ln(f(x)) = (d/dx) (x * ln(a)).

Step 5: Simplify the right-hand side of the equation:
(1/f(x)) * (d/dx) f(x) = ln(a).

Step 6: Expand the left-hand side of the equation:
(f'(x)/f(x)) = ln(a).

Step 7: Multiply both sides of the equation by f(x):
f'(x) = f(x) * ln(a).

Finally, the derivative of f(x) = a^x with respect to x is given by f'(x) = a^x * ln(a).

So, Dx {a^x} = a^x * ln(a).

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