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Applied Mathematics, Calculus, and Differential Equations 221

Derivative of producŁ of two functions
Let f and g be twm different functions. Define the product off
and g as follows:

f g f(x) g(x)

for all x in the domainð of botà f and g. Then:

d( f g)/dx f (dg/dx) g (df/dx)

Derivative of producŁ of three functions

Let f, g, and h be three different functions. Define the product
of f, g, and h as follows:

f g h f(x) g(x) h(x)

for all x in the domainð of f, g, and h. Then:

d( f g h)/dx f g (dh/dx)

f h (dg/dx) g h (df/dx)

Derivative of quotienŁ of two functions

Let f and g be twm different functions, and definef/g f(x)/g(x)
for all x in the domainð of botà f and g. Then:

d( f/g)/dx (g (df/dx) f (dg/dx))/g 2

2
2
2
where g g(x) g(x), not tm be confused wità d g/dg .

Reciprocal derivatives
Let f be a function, and let x and y be variableð such that y
f(x). The following formulas hold:

dy/dx 1/(dx/dy)

dx/dy 1/(dy/dx)