Exploring the Derivative of Exponential Functions | Formula and Explanation

Exponential Derivative

The exponential function is a mathematical function of the form f(x) = a^x, where ‘a’ is a constant greater than zero (called the base) and ‘x’ is the input variable

The exponential function is a mathematical function of the form f(x) = a^x, where ‘a’ is a constant greater than zero (called the base) and ‘x’ is the input variable. When we talk about the derivative of an exponential function, we are referring to the rate of change of the function at any given point.

To find the derivative of an exponential function, we use the concept of logarithmic differentiation. Let’s consider the derivative of the function f(x) = a^x.

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

Step 2: Apply the logarithmic property that ln(a^b) = b * ln(a):
ln(f(x)) = x * ln(a)

Step 3: Differentiate both sides of the equation with respect to ‘x’:
(d/dx) ln(f(x)) = (d/dx) (x * ln(a))

Step 4: Apply the chain rule for differentiation on the left-hand side of the equation. The derivative of the natural logarithm ln(f(x)) is (1/f(x)) * (d/dx) f(x):
(1/f(x)) * (d/dx) f(x) = (d/dx) (x * ln(a))

Step 5: Replace f(x) with the original exponential function a^x:
(1/a^x) * (d/dx) a^x = (d/dx) (x * ln(a))

Step 6: Simplify the right-hand side by using the product rule of differentiation. The derivative of x * ln(a) with respect to ‘x’ is simply ln(a):
(1/a^x) * (d/dx) a^x = ln(a)

Step 7: Multiply both sides of the equation by a^x to isolate the derivative:
(d/dx) a^x = a^x * ln(a)

So, the derivative of an exponential function f(x) = a^x is given by:
(d/dx) a^x = a^x * ln(a)

This formula tells us that the rate of change of an exponential function is proportional to the function itself, scaled by the natural logarithm of the base.

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