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Multiplication by -1

Remember that putting a negative sign in front of a function means the same thing as multiplying that function by -1.

Sample Question

Let g(x) = -x2. Then we could think of this function as

g(x) = (-1)(x2),


g ' (x) = (-1)(x2)' = (-1)(2x) = -2x.

The moral of the story is that the derivative of the negative of f is the negative of the derivative of f:

(-f(x))' = -f ' (x).

Sample Problem

Let f(x) = -sin x. Then 

f ' (x) = -(sin x)' = -cos(x).

This is really just a special case of the constant multiple rule, given that -1 is a constant and all.

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