The Quotient Rule

The next rule of differentiation to discuss will be the quotient rule.  Now that we have already learned the product rule, this rule will be a lot easier but strict.  Here is the formula:

Given that f(x) = g(x) / h(x) ,

f '(x) = [h(x) * g'(x)] - [g(x) * h'(x)] / [h(x)]²

To simply put this into words, it would go something like this:  "the derivative of f(x) is the derivative of the bottom times the top, minus the top times the derivative of the bottom, over the bottom squared."  At first glance, it looks very similar to the product rule.  However, you must be very careful when distinguishing the two.  The quotient rule has SUBTRACTION in its numerator, so the ORDER MATTERS.  Messing this up while immediately give you an incorrect answer.  Other than keeping your rule straight, however, this rule is quite simple.  Lets look at an example:

f(x) = x² / 4x

then

f '(x) = (4x)(2x) - (x²)(4) / (4x)^2

f '(x) = 8x² - 4x² / 16x²

f '(x) = 4x² / 16x²

f '(x) = 4 / 16

f '(x) = 1/4

Thanks for reading and please ask questions!