Product Rule: Mixed Concepts & Common Pitfalls

The derivation of the Product Rule begins with the limit definition of the derivative applied to the product function f(x)=u(x)v(x). This gives us option A, which can then be manipulated algebraically (by adding and subtracting u(x+h)v(x)) to eventually arrive at the Product Rule formula f'(x)=u'(x)v(x)+u(x)v'(x).

C'(x) represents the derivative of the total cost function with respect to quantity x. In economics, this is called the marginal cost, which represents the additional cost of producing one more unit. It tells us how quickly the total cost is changing as production levels change.

The Product Rule is f'(x) = u'(x)v(x) + u(x)v'(x), not minus. Option C shows a subtraction, which would be incorrect. This is a common sign error - students sometimes confuse the Product Rule with the Quotient Rule, which has a subtraction in the numerator. 

The Product Rule states that f'(x) = u'(x)v(x) + u(x)v'(x), NOT u'(x)v'(x). This is a common misconception. Option C is false because the derivative of a product is not simply the product of the derivatives. For example, if u(x) = x and v(x) = x, then f(x) = x², f'(x) = 2x, but u'(x)v'(x) = 1 × 1 = 1, which is incorrect.

The correct extended Product Rule for three functions is p'(x) = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x). The student made two errors:

1. They forgot the middle term f(x)g'(x)h(x)

2. They incorrectly multiplied derivatives in the first term (f'(x)g'(x)h(x)) instead of multiplying each derivative with the original functions of the others.