3 mistakes learners make with the chain rule, each one named and explained.
Differentiate the outside, leave the inside alone, then multiply by the derivative of the inside.
(2x + 1)^5 differentiates to 5(2x + 1)^4, with the derivative of the inside missing.
The outer power rule is the familiar part and it produces a complete-looking answer.
Question. (2x + 1)^5
A common answer. 5(2x + 1)^4
The answer. 10(2x + 1)^4
Why. The inside differentiates to 2, and that 2 multiplies the whole thing.
Chain rule has two factors. If your answer has only one, the chain is missing.
The derivative of the inside is applied and the outer power is not brought down.
Once the inner derivative is remembered it feels like the hard part is handled.
Question. (2x + 1)^5
A common answer. 2(2x + 1)^4
The answer. 10(2x + 1)^4
Why. Both the 5 from the power and the 2 from the inside multiply, giving 10.
Write both multipliers down before combining them. Neither is optional.
The coefficient is brought down correctly and the power stays where it was.
Bringing the power down is the step you concentrate on, and reducing it is an afterthought.
Question. (2x + 1)^5
A common answer. 10(2x + 1)^5
The answer. 10(2x + 1)^4
Why. Differentiating always lowers the power by one.
The degree of a derivative is always one less. Check it before moving on.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.