let f be the function satisfying f'(x) = x√f(x) for all real numbers where f(3) = 25 find f'(3)
I get y = 1/12 (4x^3 + 12c x^(3/2) + 9c^2)
using y(3) = 25, that gives
y = 1/9 (x^3 -6(√3±5) x^(3/2) + 252±99√3)
Looks pretty hairy, so check my math
To find the derivative of the function f'(x), we need to use the given equation f'(x) = x√f(x).
We know that f'(x) represents the derivative of f(x), or the rate of change of f(x) with respect to x.
The equation f'(x) = x√f(x) tells us that the derivative of f(x) at any point x is equal to x times the square root of f(x).
To find f'(3), we can substitute x = 3 into the equation f'(x) = x√f(x).
f'(3) = 3√f(3)
Now we need to find the value of f(3). Given f(3) = 25, we substitute this value into the equation:
f'(3) = 3√25
Simplifying the square root of 25:
f'(3) = 3 * 5
f'(3) = 15
Therefore, f'(3) = 15.