How to Find the Derivative of b^x: A Guide to Logarithmic Differentiation

Derivative of b^x

To find the derivative of b^x, where b is a constant, you can use the concept of logarithmic differentiation

To find the derivative of b^x, where b is a constant, you can use the concept of logarithmic differentiation.

Let y = b^x.

To find dy/dx, take the natural logarithm of both sides:

ln(y) = ln(b^x)

Using the logarithmic property, ln(b^x) = x * ln(b):

ln(y) = x * ln(b)

Now, differentiate both sides with respect to x:

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

Using the chain rule, the derivative of ln(y) with respect to x is (1/y) * dy/dx:

(1/y) * dy/dx = ln(b)

Multiplying both sides by y:

dy/dx = y * ln(b)

Since y = b^x, substitute it back into the equation:

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

So, the derivative of b^x with respect to x is b^x times the natural logarithm of b: dy/dx = b^x * ln(b).

More Answers: