the.com/product rule
Differentiate a team sport: one factor runs while the other holds still, then switch.
means The calculus rule for differentiating a product of two functions: the derivative is the first function times the derivative of the second, plus the second function times the derivative of the first.
from Not a word with a colorful backstory but a mathematical result — it falls out of the definition of the derivative as a limit. Expand (f(x+h)g(x+h) − f(x)g(x))/h, cleverly add and subtract f(x+h)g(x) in the numerator, split the fraction in two, and let h shrink to zero: one piece becomes f times g's derivative, the other becomes g times f's derivative. The name itself is plain and functional, dating to when calculus textbooks began cataloguing differentiation techniques — 'product rule' simply because it tells you what to do with a product.
- formula
- (fg)' = f'g + fg'
- common mistake
- multiplying the derivatives together instead
- extends to
- three or more factors, one term each
- cousin rule
- quotient rule is just this plus subtraction