Rules for Derivatives and Chain Rule Counterparts

Original Rule	Generalized Rule (Chain Rule)
d dx f(x) = g(x)	d dx f(u) = g(u) du dx	General form of Chain Rule
d dx xn = nx n1	d dx un = nun1 du dx	Generalized Power Rule
d dx 4x1 = 4x2	d dx 4u1 = 4u2 du dx	Example
d dx sin x = cos x	d dx sin u = cos u du dx	(Press herefor text on trig functions.)
d dx logb(x) = 1 x ln(b)	d dx logb(u) = 1 u ln(b) du dx	(Logarithm with arbitary base)
d dx ln x = 1 x	d dx ln (u) = 1 u du dx	(Natural Logarithm)
The Above Rule in Words:The derivative of the natural logarithm of a quantity is the reciprocal of that quantity, times the derivative of that quantity.
d dx bx = bx ln(b)	d dx bu = bu ln(b) du dx
d dx ex = ex	d dx eu = eu du dx
The Above Rule in Words:The derivative of e raised to a quantity is e raised to that quantity, times the derivative of that quantity.

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