Rules for Derivatives and Chain Rule Counterparts

Original Rule	Generalized Rule (Chain Rule)	Comments
d dx f(x) = g(x)	d dx f(u) = g(u) du dx	General form of Chain Rule
d dx xn = nx n-1	d dx un = nun-1 du dx	Generalized Power Rule
d dx 4x-1/2 = -2x-3/2	d dx 4u-1/2 = -2u-3/2 du dx	An example of the above rule
d dx sin x = cos x	d dx sin u = cos u du dx	Take me to 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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