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| === Properties === Derivatives are linear: given a function defined like ''f(x) = αg(x) + βh(x)'', {{attachment:sum.svg}}. The '''product rule''' states that, given a function defined like ''f(x) = g(x)h(x)'', {{attachment:prod.svg}}. This follows from the [[Calculus/Differential|differential]]; substitute ''g'' and ''h'' for ''g(x)'' and ''h(x)'': ''f = gh'' ''df = f,,g,,dg + f,,h,,dh'' ''df/dx = f,,g,,(dg/dx) + f,,h,,(dh/dx)'' And clearly the partial derivatives ''f,,g,,'' and ''f,,h,,'' are equal to ''h'' and ''g'' respectively, giving: ''df/dx = h(dg/dx) + g(dh/dx)'' Substituting back in the original functions gives the product rule. Because the derivative of a constant is 0, the product rule also proves that ''(αf)' = αf' ''. |
Derivative
A derivative is an instantaneous rate of change with respect to an input variable.
Contents
Rules
, for a > 0
, for x > 0
, for x > 0 and a > 0
See the trigonometric functions' defined here.
, for -1 < x < 1
, for -1 < x < 1
Properties
Derivatives are linear: given a function defined like f(x) = αg(x) + βh(x), 
.
The product rule states that, given a function defined like f(x) = g(x)h(x), 
. This follows from the differential; substitute g and h for g(x) and h(x):
f = gh
df = fgdg + fhdh
df/dx = fg(dg/dx) + fh(dh/dx)
And clearly the partial derivatives fg and fh are equal to h and g respectively, giving:
df/dx = h(dg/dx) + g(dh/dx)
Substituting back in the original functions gives the product rule.
Because the derivative of a constant is 0, the product rule also proves that (αf)' = αf' .