Calculating the Derivative of the Exponential Function with Base a Raised to the Power of x – Chain Rule Method

dy/dx a^x

The expression “dy/dx a^x” represents the derivative of the exponential function with base “a” raised to the power of “x”

The expression “dy/dx a^x” represents the derivative of the exponential function with base “a” raised to the power of “x”. To find this derivative, we can use the chain rule.

Let’s start by rewriting the function in terms of e, the natural logarithm base, since it will simplify the derivative calculation. We know that a^x = e^(x * ln(a)).

Now, we can proceed with finding the derivative of this function:

dy/dx = d(e^(x * ln(a))) / dx

Using the chain rule, we can write this as:

dy/dx = (d(e^(x * ln(a)))) / d(x * ln(a)) * d(x * ln(a)) / dx

The first part, d(e^(x * ln(a))) / d(x * ln(a)) is the derivative of e^u, where u = x * ln(a). This can be found using the exponential rule, which states that the derivative of e^u with respect to u is simply e^u. Thus,

d(e^(x * ln(a))) / d(x * ln(a)) = e^(x * ln(a))

The second part, d(x * ln(a)) / dx, is the derivative of the product of two functions. Using the product rule, we have:

d(x * ln(a)) / dx = x * d(ln(a)) / dx + ln(a) * d(x) / dx

Since d(ln(a)) / dx and d(x) / dx are both equal to 1, we can simplify this to:

d(x * ln(a)) / dx = x * 1 + ln(a) * 1 = x + ln(a)

Substituting these values back into the original equation, we have:

dy/dx = e^(x * ln(a)) * (x + ln(a))

Therefore, the derivative of a^x (or e^(x * ln(a))) is e^(x * ln(a)) times the sum of x and ln(a):

dy/dx = a^x * (x + ln(a))

This derivative represents the rate of change of the exponential function with respect to x at any given point.

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