Derivative of f(x) = a^x with Respect to 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 logarithmic differentiation

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

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

Step 2: Apply the properties of logarithms to simplify the equation. Using the rule ln(a^x) = x ln(a), we get ln(f(x)) = x ln(a).

Step 3: Differentiate both sides of the equation with respect to x. The derivative of ln(f(x)) with respect to x is 1/f(x) * f'(x), while the derivative of x ln(a) is ln(a). Therefore, we have 1/f(x) * f'(x) = ln(a).

Step 4: Solve for f'(x). Multiply both sides of the equation by f(x) to get f'(x) = f(x) * ln(a).

Since f(x) = a^x, we have f'(x) = a^x * ln(a).

Therefore, the derivative of a^x with respect to x is a^x * ln(a).

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