instantaneous gradient at a specific point on a curve defined?","back":"The gradient of the tangent at that point.","points":1},{"id":"48d045df-5031-4c28-8d7b-ec960967593e","cardType":"cloze","front":"The derivative from first principles is defined by the formula: $f'(x) = \\lim_{h \\to 0} \\text{[...]}$","back":"The derivative from first principles is defined by the formula: $f'(x) = \\lim_{h \\to 0} \\boxed{\\frac{f(x+h) - f(x)}{h}}$","points":null},{"id":"2a51b3db-a375-4ac0-94bf-f36dc355c98a","cardType":"basic","front":"What does the symbol $\\lim_{h \\to 0}$ represent in the first principles formula?","back":"The value the expression approaches as $h$ gets closer and closer to 0.","points":1},{"id":"72178340-0606-4335-a251-4e8c0e1cb06e","cardType":"basic","front":"Why is it mathematically impossible to substitute $h=0$ into the first principles formula immediately?","back":"It would result in division by zero.","points":1},{"id":"5530a422-d1b4-41f9-8666-2c32f3659b86","cardType":"cloze","front":"When expanding $(x+h)^2$ for a first principles proof, the result is $\\text{[...]}$.","back":"When expanding $(x+h)^2$ for a first principles proof, the result is $\\boxed{x^2 + 2xh + h^2}$.","points":null},{"id":"e789cf3b-9676-43ce-bd2b-347fcb91976a","cardType":"cloze","front":"The expansion for a cubic term $(x+h)^3$ is $\\text{[...]}$.","back":"The expansion for a cubic term $(x+h)^3$ is $\\boxed{x^3 + 3x^2h + 3xh^2 + h^3}$.","points":null},{"id":"9063a8bd-cc42-4ecc-82e4-995d1c29da1b","cardType":"basic","front":"In a first principles proof, what must be true of every term remaining in the numerator after subtracting $f(x)$?","back":"Every term must contain a factor of $h$.","points":1},{"id":"4716b12d-05fa-4790-b5e5-0c422927437f","cardType":"basic","front":"What algebraic step must occur before letting $h \\to 0$ in a first principles proof?","back":"Factorising out and cancelling $h$ from the numerator and denominator.","points":2},{"id":"b6c196f5-a2a4-4684-b3b3-b283894dcad9","cardType":"basic","front":"When proving the derivative of $3x^2$ from first principles, what is the expression for $f(x+h)$?","back":"$$3(x+h)^2 = 3x^2 + 6xh + 3h^2$","points":2},{"id":"0b71d816-bb72-40d5-aee0-69d1d0e5e536","cardType":"basic","front":"If an exam question asks to differentiate \"from first principles\", can you just use the power rule?","back":"No; the full limit definition and algebraic proof must be shown for marks.","points":1},{"id":"84e38b61-4362-4983-92b8-9722c60df62a","cardType":"basic","front":"What is the common mistake when subtracting $f(x)$ from $f(x+h)$ in complex expressions?","back":"Forgetting to subtract the whole function (not using brackets), which leads to sign errors.","points":1},{"id":"a8451e9a-74e8-42aa-b120-55b7f5a68537","cardType":"basic","front":"How does the constant $a$ behave when differentiating $f(x) = ax^n$ from first principles?","back":"It remains as a multiplier for every term in the expansion and the final derivative.","points":2},{"id":"81b0f3fe-ec55-49d6-97ff-89da83bf7865","cardType":"cloze","front":"In the context of first principles, $h$ represents a [ ] in $x$.","back":"In the context of first principles, $h$ represents a small change in $x$.","points":null},{"id":"ceae8c99-e081-4854-9f9e-60359a096d6c","cardType":"basic","front":"Outline the final three steps of the core method for differentiation from first principles.","back":"

Simplify the numerator
Cancel the factor of $h$
Let $h \\to 0$

","points":3},{"id":"0285c751-def9-4750-87a8-f52145112ec4","cardType":"basic","front":"Why is it useful to use the power rule when answering a first principles question?","back":"To check your final answer<

12.2 Differentiation from first principles Flashcards

12.2 Differentiation from first principles Flashcards

12.2 Differentiation from first principles Flashcards

12.2 Differentiation from first principles Flashcards