Quotient Rule Differentiation
Use the quotient rule when a function is written as one differentiable expression divided by another:
a(x) / b(x)
The derivative is found by differentiating both parts and combining them in a specific pattern.
1) Identify the numerator and denominator
- Let the top be u(x)
- Let the bottom be v(x)
2) Apply the quotient rule
d/dx [u/v] = (v·u' - u·v') / v²
This means:
- differentiate the numerator to get u'
- differentiate the denominator to get v'
- multiply v by u'
- subtract u times v'
- divide everything by v²
3) Simplify carefully
After differentiating:
- combine like terms if possible
- reduce factors if the expression allows it
- write the final answer in a clean simplified form
4) Check your work
A good check is to make sure:
- the denominator is squared
- the order is v·u' - u·v', not the reverse
- any sign changes are handled correctly
- the final expression is simplified
For extra confidence, you can also compare your result with a product-rule rewrite of the quotient, when appropriate.