How to Find the Derivative of b^x using Logarithmic Differentiation

Derivative of b^x

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

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

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

ln(y) = ln(b^x)

Using the logarithmic property ln(a^b) = b ln(a), we can rewrite the right side of the equation as:

ln(y) = x * ln(b)

Now, 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, and the product rule on the right side:

(1/y) * dy/dx = 1 * ln(b) + x * d/dx (ln(b))

Simplifying the right side, we get:

(1/y) * dy/dx = ln(b) + x * (1/b) * db/dx

Now, let’s solve for dy/dx by multiplying both sides of the equation by y:

dy/dx = y * (ln(b) + x * (1/b) * db/dx)

But remember that y is equal to b^x:

dy/dx = b^x * (ln(b) + x * (1/b) * db/dx)

So, the derivative of b^x with respect to x is:

dy/dx = b^x * (ln(b) + x * (1/b) * db/dx)

In summary, the derivative of b^x is b^x times the natural logarithm of b, plus x times b^x times the derivative of b with respect to x.

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