Essential Knowledge 2.1C3

Essential Knowledge 2.1C3 Students will know that sums, differences, products, and quotients of functions can be differentiated using derivative rules.

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Not all functions we want to differentiate will just have single terms. These rules help us know what to do.

The thing to remember is that multiplication and division have special rules. This is the first time all year this has happened (limits did not have special rules for multiplication and division). So just remember, this is just for derivatives, not limits.

Sums and Differences

Sums and differences have to special rules. Just differentiate each term, as if you were distributing the derivative

Example

Products

Products have a special rule, and it looks like this

mnemonic : "1 d 2 plus 2 d 1"

This means you need to pick a function to be f(x), pick one to be g(x), then use the formula. An example will make this more clear.

Example

First of all, get that 4 out of there to make things simpler

Now we need to pick a part that will be f(x) and a part that will be g(x). This one's pretty obvious, we have a x³ and a cos(x). Let's let the x³ be f(x) and the cos(x) be g(x). In other words f(x) = x³ and g(x) = cos(x).

It looks like we'll need to know f '(x) and g'(x). So lets figure those out.

f(x) = x³

f'(x) = 3x²

g(x) = cos(x)

g'(x) = -sin(x)

Now we can use the formula

Quotients

Quotients have a special rule, and it looks like this

mnemonic : "low d high minus high d low over the square of what's below"

This means you need to pick a function to be f(x), pick one to be g(x), then use the formula. An example will make this more clear.

Example

The numerator must be f(x) and the denominator must be g(x). This means f(x) = tan(x) and g(x) = x⁴.

It looks like we'll need to know f '(x) and g'(x). So lets figure those out.

f(x) = tan(x)

f'(x) = sec²(x)

g(x) = x⁴

g'(x) = 4x³

Now we can use the formula