If f(x) = 3x^(ln(x)), find f'(10)
To find the derivative of the function f(x) = 3x^(ln(x)) and evaluate it at x = 10, we can use the chain rule.
The chain rule states that if we have a function of the form g(h(x)), then its derivative is given by g'(h(x)) * h'(x), where g'(x) is the derivative of g(x) and h'(x) is the derivative of h(x).
Let's break down the function f(x) = 3x^(ln(x)) into two parts:
g(x) = 3x^y, where y = ln(x)
h(x) = ln(x)
Now, let's find the derivatives of g(x) and h(x):
g'(x) = 3x^(y-1) * (1 * y'), where y' represents the derivative of y with respect to x.
h'(x) = 1/x
Now, let's find the value of y' by taking the derivative of y = ln(x) using the chain rule:
y' = 1/x
Now, substitute these values back into g'(x):
g'(x) = 3x^(y-1) * (1/x) = 3x^(ln(x)-1) * (1/x) = 3 * (x^(ln(x)-1) / x)
Finally, to find f'(10), substitute x = 10 into g'(x):
f'(10) = 3 * (10^(ln(10)-1) / 10)
Now, you can use a calculator or math software to calculate the numerical value of f'(10).