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Home Differentiation Differentials and Tangent Lines Changes of y Along the Curve  
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Changes of y Along the Curve
Given a curve y = f (x), suppose that x starts out with the value a and then changes by an infinitesimal amount Δx. What happens to y? Along the curve, y will change by the amount f(a + Δx)  f(a) = Δy. But along the tangent line y will change by the amount l(a + Δx)  l(a) = [f'(a)(a + Δx  a) + b]  [f'(a)(a  a) + b] = f'(a) Δx. When x changes from a to a + Δx, we see that: change in y along curve = f(a + Δx)  f(a), change in y along tangent line = f'(a) Δx. In the last section we introduced the dependent variable Δy, the increment of y, with the equation Δy = f (x + Δx)  f(x). Δy is equal to the change in y along the curve as x changes to x + Δx. The following theorem gives a simple but useful formula for the increment Δy.


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