Florida B.E.S.T. MA.912.C.2.3
The Standard
Prove the rules for finding derivatives of constants, sums, products, quotients and the Chain Rule.
Florida B.E.S.T. Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students use the limit definition, algebra, and limit laws to justify derivative formulas for combinations and compositions of functions. They explain each transformation and the assumptions needed, rather than verifying formulas with examples.
What Mastery Looks Like
- A student can begin with the limit definition and produce valid general proofs for each derivative rule. They insert useful zero terms or factors, apply limit laws, and state conditions such as a nonzero denominator. They can explain how the chain proof links outer and inner rates.
Common Misconceptions
- Students often treat several examples as a proof or use the target rule inside its own proof. Common errors include c'=1, (fg)'=f'g', (f/g)'=f'/g', and a missing inner derivative. In chain-rule proofs, they may divide by an inner-function increment without handling when it equals zero.
How to Assess It
- Give this exit ticket: Prove the product rule from the limit definition by adding and subtracting f(x+h)g(x), then label each limit law used.
Lesson moves
Ways to Teach It
Give groups shuffled cards showing the product-rule limit proof; students arrange them, then justify each algebra and limit step.
Ask students to explain why checking ten examples cannot prove the quotient rule for all differentiable functions.
Run a proof relay where teams complete missing steps in constant, sum, quotient, and chain-rule derivations.
Model a growing circle with A(t)=πr(t)^2, then derive its rate formula and identify where the chain rule enters.
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Related Standards
- MA.912.C.1.2
Determine the value of a limit if it exists algebraically using limits of sums, differences, products, quotients and compositions of continuous functions.
- MA.912.C.2.5
Find the derivatives of implicitly defined functions.
- MA.912.C.2
Develop an understanding for and determine derivatives.
- MA.912.C.2.4
Apply the rules for finding derivatives of constants, sums, products, quotients and the Chain Rule to solve problems with functions limited to algebraic, trigon...
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