Understanding Basic Rules of Calculus

Product, Quotient & Chain Rules

Ever wondered how to differentiate two functions that are stuck together? The product rule is your go-to when you need to find the derivative of two multiplied functions like u$x$ · v$x$.

The formula is surprisingly straightforward: first × (derivative of second) + second × (derivative of first). So if you have y = $3x² + 5x$(6x³), you'd identify u = 3x² + 5x and v = 6x³, then find their derivatives u' = 6x + 5 and v' = 18x².

The quotient rule handles division between functions using the pattern: bottom × (derivative of top) - top × (derivative of bottom), all divided by (bottom)². For fractions like $7x² - 3x$/$2x³ + 4x²$, this rule keeps everything organised and manageable.

Quick Tip: Remember "bottom d-top minus top d-bottom over bottom squared" - it's easier to memorise than the formal notation!

The chain rule comes into play when you've got functions nested inside each other, like $3x² - 4x + 7$⁴. You differentiate the outer function first, then multiply by the derivative of the inner function. Think of it as peeling an onion - work from the outside in!