Math Explained: Finding the Derivative of b^x Using Logarithmic Differentiation

deriv of b^x

To find the derivative of b^x, where b is a constant and x is a variable, we can use the properties of logarithmic differentiation

To find the derivative of b^x, where b is a constant and x is a variable, we can use the properties of logarithmic differentiation.

Let’s start by expressing b^x as e^(ln(b^x)), where e is the base of the natural logarithm and ln represents the natural logarithm.

Using the properties of logarithms, we can rewrite ln(b^x) as x * ln(b).

Now, we can express b^x as e^(x * ln(b)).

To find the derivative of e^(x * ln(b)), we can use the chain rule. The chain rule states that if we have a composite function f(g(x)), the derivative of f(g(x)) is f'(g(x)) * g'(x).

In this case, g(x) = x * ln(b), and f(g) = e^g. Using the chain rule, we have:

f'(g) = d/dg(e^g) = e^g

g'(x) = d/dx(x * ln(b)) = ln(b)

Therefore, the derivative of b^x is:

d/dx(b^x) = f'(g) * g'(x) = e^(x * ln(b)) * ln(b)

Simplifying this expression further, we have:

d/dx(b^x) = b^x * ln(b)

So, the derivative of b^x with respect to x is b^x times ln(b).

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