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Find, from the first principles, the gradient function of:

f(x)=x


f'(x)= \lim_{h \to \ 0 (f(x+h)-f(x))/(h) }

User Nishantv
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2 Answers

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\begin{aligned}f'(x) &= \lim_(h \to 0) (f(x+h) - f(x))/(h) \\&= \lim_(h \to 0) ((x+h) - x)/(h) \\&=\lim_(h \to 0) (h)/(h) \\&=\lim_(h \to 0) (1) && (\text{\footnotesize since $h/h = 1$ for $h\\e 0$})\\&= 1\end{aligned}

h/h = 1 is valid for h ≠ 0. The limit does not care about h at 0; it only cares about the values around it.

User Osa
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3 votes
f(x+h) = x+h, and f(x) = x

So, your limit is:


\lim_(h \to 0) (x+h - x)/(h) = \lim_(h \to 0) (h)/(h)

Applying l'hopital's rule,


\lim_(h \to 0) (h)/(h) = \lim_(h \to 0) (1)/(1) = 1

Giving a gradient of 1.
User Renm
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