Exponential Derivative
The derivative of an exponential function is a fundamental concept in calculus
The derivative of an exponential function is a fundamental concept in calculus. To find the derivative of an exponential function, we use the basic rules of differentiation.
Let’s consider the general form of an exponential function:
f(x) = a * b^x
where “a” and “b” are constants.
The base, “b”, is usually a positive number greater than 1.
To find the derivative of this function, we differentiate with respect to “x” using the following steps:
Step 1: Start with the general form of the exponential function.
f(x) = a * b^x
Step 2: Take the natural logarithm of both sides of the equation.
ln(f(x)) = ln(a * b^x)
Step 3: Apply the logarithmic rule for product of two factors.
ln(f(x)) = ln(a) + ln(b^x)
Step 4: Apply the logarithmic rule for exponent.
ln(f(x)) = ln(a) + x * ln(b)
Step 5: Differentiate both sides of the equation with respect to “x”.
(d/dx) ln(f(x)) = (d/dx) [ln(a) + x * ln(b)]
Step 6: Differentiate both terms on the right side of the equation using the derivative rules.
(d/dx) ln(f(x)) = 0 + ln(b) * (d/dx) x
Step 7: Simplify the equation and recall that (d/dx) x equals 1.
(d/dx) ln(f(x)) = ln(b)
Step 8: Apply the exponential function to both sides of the equation to solve for f(x).
f'(x) = b^(ln(f(x)))
Step 9: Substitute the original exponential function back into the equation.
f'(x) = b^(ln(a * b^x))
Step 10: Apply the logarithmic rule for exponent.
f'(x) = a * b^x * b^(ln(b))
Step 11: Recall that b^(ln(b)) simplifies to e, where e is the base of natural logarithm (approximately equal to 2.71828).
f'(x) = a * b^x * e
So, the derivative of the exponential function f(x) = a * b^x with respect to x is f'(x) = a * b^x * e.
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