Answer: To differentiate a function using the first principle, we first need to know the function and it's original value. Given that the function is f(x) = 5√x .
The first principle of differentiation states that the derivative of a function f(x) at a point x is approximately equal to the change in f(x) for an infinitesimal change in x. We can use this principle to find the derivative of a function f(x) at a point x by calculating the limit of the ratio (f(x+dx)-f(x))/dx as dx approaches zero.
To differentiate f(x) = 5√x with respect to x using the first principle, we take the following steps:
Find the change in f(x) as x changes by dx, which is f(x + dx) - f(x)
Divide that change by dx to find the slope of the function at that point
Take the limit of that slope as dx approaches zero
So:
f(x + dx) = 5√(x + dx)
f(x) = 5√x
change in f(x) = f(x + dx) - f(x) = 5√(x + dx) - 5√x
slope = change in f(x) / dx
= (5√(x + dx) - 5√x) / dx
The limit as dx approaches zero is the derivative of the function,
lim (dx->0) [(5√(x + dx) - 5√x) / dx]
= lim (dx->0) [(5(x+dx)^1/2 - 5x^1/2) / dx]
= 5 * lim (dx->0) [(x+dx)^1/2 - x^1/2] / dx
= 5 * (1/2)x^(-1/2)
So the derivative of the function f(x) = 5√x is (5/2)x^(-1/2)
This is the slope of the function at a point x, which is the rate of change of the function at that point x.
Explanation: