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Intermediate Mathematics - Linear combinations of functions

Linear combinations of functions

A_ linear combination_ of functions is formed when constant multiples of
the functions are added together.

For example, f(x) = sin x + x is a linear combination of the two functions
u(x) = sin x and v(x) = x.

In general, if u(x) and v(x) are two functions of_ x,_ then a linear
combination is

f(x) = a(u(x)) + b(v(x))

= au+bv, where _a_ and_ b_ are real constants.

The derivative of this linear combination is the linear combination of the
derivatives:

f'(x) = a(u'(x)) + b(v'(x))

= au' + bv'

This sounds complicated but it is common sense and something you will do
all the time without thinking about it.

The derivative of f(x) = 2x - sin(x) + 3e _ is_

f'(x) = 8x - cos(x) + 3e

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