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Definition : A function f ( x ) is called differentiable at x = a if f ′ ( x ) exists & f ( x ) is called differentiable onto an interval if the derivative present for each of the point in that interval.
The next theorem illustrates us a very nice relationship among functions which are continuous & those that are differentiable.
Theorem: If f ( x ) is differentiable at x = a then f ( x ) is continuous at the point x = a.
Note as well that this theorem does not work in reverse order . Assume f ( x ) = |x| and take a look at,
So,f ( x ) = xis continuous at x =0 to sketch the graph of a function.
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Theorem, from Definition of Derivative If f(x) is differentiable at x = a then f(x) is continuous at x =a. Proof : Since f(x) is differentiable at x = a we know, f'(a
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Below is the sketch of a function f ( x ) . Sketch the graph of the derivative of this function f ′ ( x ) . Solution : At first glance it seems to an all however impossib
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