Final answer:
To find the derivative of the function f(x) = 3x^(ln(x)), we can use the chain rule and logarithmic differentiation. Differentiating the function, we get f'(x) = (ln(x) + 1)(3x^(ln(x)))/(x ln(3x)). By substituting x = 10, we find that f'(10) ≈ 14.52.
Step-by-step explanation:
To find the derivative of the function f(x) = 3x^(ln(x)), we can use the chain rule. Let's start by differentiating the outer function, which is the constant 3, this will give us 0. Then, we differentiate the inner function, which is x^(ln(x)).
To differentiate x^(ln(x)), we will use logarithmic differentiation. Taking the natural logarithm ln of both sides gives us ln(f(x)) = ln(3x^(ln(x))). Applying the logarithmic property ln(ab) = b ln(a), we get ln(f(x)) = (ln(x))(ln(3x)). Differentiating implicitly, we get f'(x)/f(x) = (ln(x))(1/ln(3x)) + ln(3x)(1/x). Rearranging the equation, we get f'(x) = (ln(x) + 1)(3x^(ln(x)))/(x ln(3x)).
To find f'(10), we substitute x = 10 into the derivative expression. So, f'(10) = [(ln(10) + 1)(3(10^(ln(10))))]/[10 ln(3(10))]. Evaluating the logarithms and performing the calculations, f'(10) ≈ 14.52.