Derivative of b^x: Learn How to Find the Derivative of b^x using the Power Rule and Natural Logarithms

Derivative of b^x

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

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

The power rule states that if we have a function of the form f(x) = c * x^n, where c is a constant and n is any real number, the derivative of f(x) is given by f'(x) = c * n * x^(n-1).

In this case, we have f(x) = b^x, where b is a constant raised to the power of x. To find the derivative, we can take the natural logarithm (ln) of both sides of the equation:

ln(f(x)) = ln(b^x)

Using the property of logarithms that ln(a^b) = b * ln(a), we can rewrite the equation as:

ln(f(x)) = x * ln(b)

Now, we can differentiate both sides of the equation with respect to x:

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

To find the derivative on the left side of the equation, we can use the chain rule. Let u = f(x), where u is the inside function, and y = ln(u), where y is the outside function. Then, the derivative of y with respect to x is given by (dy/dx) = (dy/du) * (du/dx).

Using the chain rule, we have:

(dy/dx) = (d/dx) [ln(u)]
(dy/dx) = (1/u) * (du/dx)

The derivative of ln(u) is 1/u, where u = f(x) = b^x. So, we have:

(dy/dx) = (1/f(x)) * (d/dx) [b^x]

Now, substituting f(x) = b^x and (d/dx) [b^x] = ln(b) * b^x, we get:

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

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

In summary, the derivative of b^x is ln(b).

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