Question

Let f(x) = x^(3x). Use logarithmic differentiation to determine the derivative f'(x).

Answer 1

To find the derivative of f(x) = x^(3x) using logarithmic differentiation, take the natural logarithm of both sides and differentiate.

Step-by-step explanation:

To find the derivative of f(x) = x^(3x) using logarithmic differentiation, we can take the natural logarithm of both sides of the equation:

ln(f(x)) = ln(x^(3x))

Using the logarithmic property, we can bring the exponent down as a coefficient:

ln(f(x)) = (3x)ln(x)

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

f'(x)/f(x) = 3ln(x) + 3x(1/x)

f'(x)/f(x) = 3ln(x) + 3

Multiplying both sides by f(x), we get:

f'(x) = f(x) * (3ln(x) + 3)

So, the derivative of f(x) = x^(3x) is f'(x) = x^(3x) * (3ln(x) + 3).