F(x+dx)=F(x)+F'(x)dx eqn an identity

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The equation F(x+dx)=F(x)+F'(x)dx represents a fundamental approximation derived from the definition of the derivative, where F'(x) is defined as the limit of the difference quotient as Δx approaches zero. This approximation holds true for infinitesimally small increments in x, denoted as dx. The discussion highlights the relationship between this equation and concepts such as Taylor series and the Newton-Raphson algorithm, which are essential in numerical analysis.

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F(x+dx)=F(x)+F'(x)dx

is the above eqn an identity or something?can someone explain to me what is happening in this eqn?
 
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It's the approximation which follows from the definition of the derivative

[tex]F'(x)=:\lim_{\Delta x\rightarrow 0} \frac{F(x+\Delta x)-F(x)}{\Delta x}[/tex]

by not considering the limit,but the increment in "x",viz.[itex]\Delta x[/itex],is very,very small (infinitesimal,if you prefer).

Daniel.
 
Locate a discussion of Taylor series and the Newton-Rapson algorithm. You'll find a good bit on both in a Numerical Analysis textbook. Now, if you are reading a NA text, and that makes you ask your question, then I am sorry for redirecting you to where you already are. In that case, just notice the obvious.

If dx = 0, then of course f(x+dx) = f(x) + f'(x)dx.

What happens if we move dx slightly away from zero? ...well, see dextercioby's last post.
 
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