3 mistakes learners make with the product rule, each one named and explained.
First times the derivative of the second, plus second times the derivative of the first.
One of the two terms is produced and the other never appears.
The rule is a long expression and stopping halfway still leaves something plausible.
Question. x^2 sin(x)
A common answer. x^2 cos(x)
The answer. 2x sin(x) + x^2 cos(x)
Why. Each factor takes a turn at being differentiated, giving two terms.
Write the plus sign and the second term's skeleton before you fill anything in.
The two terms are joined with a minus sign.
The quotient rule has a minus in it, and the two rules are learned together.
Question. x^2 sin(x)
A common answer. 2x sin(x) - x^2 cos(x)
The answer. 2x sin(x) + x^2 cos(x)
Why. The product rule adds its two terms; only the quotient rule subtracts.
Check which rule you are using before writing the connecting sign.
x^2 sin x differentiates to 2x cos x, both parts differentiated at once.
It is what the product rule would be if differentiation behaved like the operation it is applied to.
Question. x^2 sin(x)
A common answer. 2x cos(x)
The answer. 2x sin(x) + x^2 cos(x)
Why. The derivative of a product is not the product of the derivatives.
Two terms come out of the product rule, always. A single term means it was not used.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.