Learn about the quotient rule - when to use it and why
Say I give you a function which looks something like this:
and I ask you to take the derivative of it. What do you do?
You could for example serve me the solution on a plate, using the quotient rule:
The quotient rule
The quotient rule is just the product rule for a special case
This rule can be derived from the product rule in the following way:
We note that:
Using this fact, we write:
And that is the quotient rule again.
A small remark: in taking the derivative of we can use the chain rule: . The inner function is and the outer function is .
The most fundamental example of the quotient rule is when taking the derivative of:
As mentioned above, we can use the chain rule to find the derivative. We can also use the quotient rule.
The numerator function would then be . The derivative of is zero, so:
This example has its own name: the reciprocal rule.
Let and . Taking the derivative of the quotient, we get:
The numerator can be simplified with trigonometric rules, as . Thus, we get:
But, as we have seen when talking about trigonometric functions, . So we have just shown:
This can also be written as , if we do not use trigonometric identities when simplifying.
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