Definition: The product rule gives the derivative of two functions multiplied together: the derivative of f times g equals f prime times g plus f times g prime.
What is the rule?
If h(x) = f(x) g(x), then h'(x) = f'(x) g(x) + f(x) g'(x). You differentiate one factor at a time and add the two pieces. The derivative of a product is not the product of derivatives.
How do you apply it?
Worked example: h(x) = 3x2 ex. Take f = 3x2, so f’ = 6x, and g = ex, so g’ = ex.
Then h'(x) = 6x ex + 3x2 ex = 3x ex(2 + x).
At x = 1, h'(1) = 6e + 3e = 9e, about 24.46.
The quotient rule is a close cousin: (f/g)’ = (f’g – fg’)/g2.
What mistake is typical?
Writing f’g’ as the answer. Test with f = x and g = x: the product is x2 with derivative 2x, but f’g’ = 1. Mathbot uses this tiny test to catch the error.
Example: (3x^2 e^x)’ = 6x e^x + 3x^2 e^x