The Derivative of an Exponential Function: Step-by-Step Guide with Examples and Rules

Exponential Derivative

The derivative of an exponential function can be found using a simple rule

The derivative of an exponential function can be found using a simple rule. The general form of an exponential function is given by f(x) = a^x, where ‘a’ is a constant and ‘x’ is the variable. To find the derivative of this function, follow these steps:

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

Step 2: Apply the logarithm property that states ln(a^x) = x 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 on the left side:
(d/dx) ln(f(x)) = (d/dx) f(x) / f(x)

Step 5: Simplify the equation using the derivative ln(u) = u’ / u, where u = f(x):
f'(x) / f(x) = (d/dx) (x ln(a))

Step 6: Multiply both sides of the equation by f(x) to isolate the derivative f'(x):
f'(x) = f(x) * (d/dx) (x ln(a))

Step 7: Replace f(x) with its original form a^x:
f'(x) = a^x * (d/dx) (x ln(a))

Step 8: Simplify the derivative (d/dx) (x ln(a)) using the product rule:
f'(x) = a^x * (d/dx) x * ln(a) + x * (d/dx) ln(a)
= a^x * ln(a) + x * 0
= a^x * ln(a)

Therefore, the derivative of the exponential function f(x) = a^x is given by f'(x) = a^x * ln(a).

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