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Differential Calculus

Cool goIITian

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22 Sep 2007 10:20:07 IST

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L'hospitals rule

None

what is this L'hospitals rule and lebinetz rule?

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Priyesh

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Joined: 18 Feb 2007

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22 Sep 2007 10:37:08 IST

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l'Hôpital's rule states that for functions f(x) and g(x), if:

$lim_{x o c}f(x)=lim_{x o c}g(x)=0,$

or:

$lim_{x o c}f(x)=lim_{x o c}g(x)=pminfty,$

then:

$lim_{x o c} rac{f(x)}{g(x)} = lim_{x o c} rac{f'(x)}{g'(x)}$

where the prime (') denotes the derivative. For this rule to hold, the limit $lim_{x o c} rac{f'(x)}{g'(x)}$ must exist or else we again apply L'hopital rule again if possible.

Priyesh

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Joined: 18 Feb 2007

Posts: 1042

22 Sep 2007 10:41:09 IST

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Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an integral of the form

$\int_{y_0}^{y_1} f(x, y) \,dy$

then for $x \in (x_0, x_1)$ the derivative of this integral is thus expressible

${d\over dx}\, \int_{y_0}^{y_1} f(x, y) \,dy = \int_{y_0}^{y_1} {\partial \over \partial x} f(x,y)\,dy$

provided that

f

and $\partial f / \partial x$ are both continuous over a region in the form

$[x_0,x_1]\times[y_0,y_1].$

Alternate form

For a monovariant function

g

${d\over dx}\, \int_{f_1(x)}^{f_2(x)} g(t) \,dt = g(f_2(x)) {f_2'(x)} - g(f_1(x)) f_1'(x)$

${d\over dq}\, \int_{a(q)}^{b(q)} g(t,q) \,dt = g(b(q),q) {b'(q)} - g(a(q),q) a'(q) + \int_{a(q)}^{b(q)} {\partial \over \partial q} g(t,q) \,dt$

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