Derivative of a Quotient of Functions

Derivative of a Quotient of Functions

Division within derivatives is more complicated than the other rules we've seen so far. Make some space in the ol' memory bank for the Quotient Rule. The Quotient Rule states that the derivative of the function  is

The quotient rule is more complicated than the product rule. Here are some important bits that need to be remembered.

• The numerator involves subtraction. Not addition, but subtraction:
    
  

• The denominator is the square of the original function's denominator:
    
  

• The numerator is
    
  f 'g – fg',

not the other way around.

One way to remember this is that we read the numerator of a fraction first, and in the quotient rule we take the derivative of the numerator first:

There's also a mnemonic that may be helpful. If we think of the original function as

then the numerator in the quotient rule is

"low dee-high minus high dee-low," where "dee" means "derivative." The phrase does have a nice ring to it.

"low dee-high minus high dee-low" translates to

gf ' – fg',

but this is the same thing as

f 'g – fg'.

That's a lot of stuff to remember, but practice will make it easier. After we learn the Chain Rule we'll be able to re-create the Quotient Rule.

If a function is written as a fraction, it doesn't necessarily mean we need to use the quotient rule to find the derivative. If the denominator of a function is a constant, we can rewrite the function and avoid using the quotient rule.

• Derivatives of Those Other Trig Functions

  We found the derivatives of sine and cosine, and now that we have the quotient rule we can take derivatives of all those other trig functions we didn't discuss yet.

  We'll use three general steps to find the derivatives of these functions in the examples and exercises.

  • Rewrite the function in terms of sine and cosine.  
  • Find the derivative using the quotient rule.
  • Write the function in terms of trig functions.

  Since every trig function can be written as a ratio of sines and cosines, this method will work every time without fail. What do you think of that?