Concerns about Derivatives

Okay, I have just finished grading HW #4. This assignment had some differentiation rules. But many of you are not comprehending the purpose of a derivative rule.

For example, we had the special rule for squares of functions:

d/dx[f²(x)] = 2 f(x) f '(x)

This means that any time there is a formula squared and you need to take its derivative, you can apply this rule.

(2x+3)² corresponds to the function f(x)=2x+3 being squared. So since f '(x) = 2:

d/dx[(2x+3)²] = 2 (2x+3) (2) = 4(2x+3)

Similarly, (x²-4x+5)² corresponds to the function f(x) = x²-4x+5 being squared, and f '(x)=2x-4

d/dx[(x²-4x+5)²] = 2 (x²-4x+5) (2x-4)

You need to be an expert at identifying the form of an expression in order to apply appropriate rules of differentiation, and next semester, rules of integration (anti-differentiation).