One of the most common mistakes new calculus students make is assuming that the derivative of a product is simply the product of the derivatives. It is not.
If y = f(x) · g(x)
y' ≠ f'(x) · g'(x)
(This is FALSE!)
Because functions change at different rates, combining them requires special rules. These are the Product Rule and the Quotient Rule.
1. The Product Rule
The Product Rule is used when two functions are being multiplied together, such as x² · sin(x).
d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
Memorization Trick: "Left d-Right plus Right d-Left".
Example:
Find the derivative of y = x³ · cos(x).
- Let f = x³ (so f' = 3x²)
- Let g = cos(x) (so g' = -sin(x))
y' = 3x²cos(x) - x³sin(x)
2. The Quotient Rule
The Quotient Rule is used when one function is divided by another, such as sin(x) / x. This formula is slightly more complex and the order matters because of the subtraction sign.
d/dx [ f(x) / g(x) ] = [ g(x)f'(x) - f(x)g'(x) ] / [g(x)]²
The "Lo-d-Hi" Mnemonic
The easiest way to remember this is the famous rhyme:
- Low: The bottom function.
- d-High: Derivative of the top.
- Low Low: The bottom function squared.
Example:
Find the derivative of y = x / (x + 1).
- High = x (High' = 1)
- Low = x + 1 (Low' = 1)
y' = [ x + 1 - x ] / (x+1)²
y' = 1 / (x+1)²
3. Which rule to use?
Sometimes you can avoid the Quotient Rule by rewriting the function with a negative exponent and using the Product Rule instead. This is often faster.
Example: 1/x³ can be written as x⁻³, allowing you to use the Power Rule instead of the Quotient Rule.