To find the derivative dy/dx of the given function y = ((1+√x) * (2+√x)) / x using the increment method, we'll first calculate the derivative at a specific point, x = 1.
The increment method approximates the derivative by finding the limit of the difference quotient as h approaches 0, where h represents the increment. The formula for the increment method is:
dy/dx = lim (h -> 0) [f(x + h) - f(x)] / h
In this case, we want to find the derivative at x = 1. So, f(x) = ((1+√x) * (2+√x)) / x, and x = 1.
Now, let's compute the derivative using the increment method with x = 1:
dy/dx = lim (h -> 0) [((1+√(1+h)) * (2+√(1+h))) / (1+h) - ((1+√1) * (2+√1)) / 1] / h
Let's simplify the expression:
dy/dx = lim (h -> 0) [((1+√(1+h)) * (2+√(1+h))) - (1 * (2+1))] / h
Now, let's calculate the limit as h approaches 0. We'll use L'Hôpital's Rule, which states that if you have an indeterminate form (0/0), you can take the derivative of the numerator and denominator separately:
dy/dx = lim (h -> 0) [d/dh((1+√(1+h)) * (2+√(1+h))) - d/dh(3)] / d/dh(h)
First, find the derivative of the numerator:
dy/dx = lim (h -> 0) [(d/dh((1+√(1+h)) * (2+√(1+h))) - 0] / 1
Now, calculate the derivative of the numerator using the chain rule:
dy/dx = lim (h -> 0) [(d/dh(1+√(1+h))) * (2+√(1+h)) + (1+√(1+h)) * (d/dh(2+√(1+h)))] / 1
dy/dx = lim (h -> 0) [(0.5/√(1+h)) * (2+√(1+h)) + (1+√(1+h)) * (0.5/√(1+h))] / 1
Now, substitute h = 0:
dy/dx = [(0.5/√(1+0)) * (2+√(1+0)) + (1+√(1+0)) * (0.5/√(1+0))] / 1
dy/dx = [(0.5/√1) * (2+√1) + (1+√1) * (0.5/√1)] / 1
dy/dx = [0.5 * (2+1) + (1+1) * 0.5] / 1
dy/dx = (1.5 + 1) /
dy/dx = 2.5
So, the derivative of the given function at x = 1 using the increment method is dy/dx = 2.5.
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