Finding the Derivative of b^x | A Guide to Logarithmic Differentiation

Derivative of b^x

The derivative of b^x, where b is a constant and x is the variable, can be found using logarithmic differentiation

The derivative of b^x, where b is a constant and x is the variable, can be found using logarithmic differentiation.

Let’s start by taking the natural logarithm of both sides of the equation:

ln(y) = ln(b^x)

Using the power rule of logarithms, we can bring the exponent x down as a coefficient:

ln(y) = x * ln(b)

Now, let’s differentiate both sides of the equation with respect to x:

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

Using the chain rule on the left side, we obtain:

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

To find dy/dx, the derivative of y with respect to x, we can multiply both sides of the equation by y:

dy/dx = ln(b) * y

Now, y is equal to b^x, so we can substitute it back in:

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

Therefore, the derivative of b^x is ln(b) times b^x:

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

This is the formula you can use to find the derivative of b^x with respect to x.

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