If F is the composition of two differentiable functions fleft(u ight) and u=gleft(x ight), that is, if Fleft(x ight)=fleft(gleft(x ight) ight), then the derivative of F is the derivative of f with respect khổng lồ u times the derivative of g with respect khổng lồ x, that is, fracmathrmdmathrmdx(F)left(x ight)=fracmathrmdmathrmdx(f)left(gleft(x ight) ight)fracmathrmdmathrmdx(g)left(x ight).

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The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^n is nax^n-1. Sometimes, when using L'Hôpital's Rule, you still each an indeterminate form. In these cases, you have khổng lồ repeat the process however many times needed until you reach a form which can be evaluated. ...
If your program can vày complex cube-roots, then (cos x+isin x)^1/3=cos(x/3)+isin(x/3) Unfortunately, the standard, Cardano solution to lớn the cubic involves complex cube-roots whenever the three ...

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https://math.stackexchange.com/questions/2021092/analyzing-the-continuity-and-derivability-of-frac1-cosxx-at-0
You did well for establishing continuity at 0. Every time a function g (defined over an open interval (a,b), for simplicity) is differentiable at a point x_0in(a,b), the function f(x)=egincases dfracg(x)-g(x_0)x-x_0 & ext{if ...
https://math.stackexchange.com/questions/1975091/prove-limit-as-x-goes-to-infinity-of-cos-x-x-0-using-epsilon-delta
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