-abbb-160efa105e47","title":"1.7.7 Differentiation from first principles for powers of x Flashcards","topicId":"9fd49e24-458d-436f-90fc-af21dc50c5b5","topicSlug":"differentiation-from-first-principles-for-powers-of-x","cards":[{"id":"605ebddd-8d0c-46d9-9fe1-80ad48a204ac","cardType":"cloze","front":"For two points on $y=f(x)$ separated horizontally by $h$, the chord gradient is $\\text{[...]}$.","back":"For two points on $y=f(x)$ separated horizontally by $h$, the chord gradient is $\\boxed{\\frac{f(x+h)-f(x)}{h}}$.","points":null,"mcqOptions":null},{"id":"cfdedf71-a518-4a65-82b7-818e0e48551b","cardType":"basic","front":"What does $\\frac{f(x+h)-f(x)}{h}$ represent geometrically?","back":"The gradient of the chord between the two points.","points":1,"mcqOptions":null},{"id":"7797a4e8-c497-4028-807b-a6b0e25c60bf","cardType":"basic","front":"As $h\\to 0$, what does the chord approach?","back":"The tangent at the point.","points":1,"mcqOptions":null},{"id":"a2254850-0fb5-46e2-ac66-de7e0b810316","cardType":"basic","front":"Why does the tangent give the curve's gradient at one point?","back":"It has the same instantaneous gradient as the curve there.","points":2,"mcqOptions":null},{"id":"bbb36020-f24d-4790-80e0-208e8ded82c9","cardType":"cloze","front":"The derivative from first principles is $\\text{[...]}$.","back":"The derivative from first principles is $\\boxed{f'(x)=\\lim_{h\\to 0}\\frac{f(x+h)-f(x)}{h}}$.","points":null,"mcqOptions":null},{"id":"2eb71dd3-150a-4f4b-b095-ab3decdc78fa","cardType":"basic","front":"Why must $h\\neq 0$ when forming the difference quotient?","back":"Otherwise its denominator is zero.","points":2,"mcqOptions":null},{"id":"c8db0bac-811a-43e2-a399-7569e8d1561f","cardType":"basic","front":"Why is $h=0$ not substituted before simplifying the difference quotient?","back":"It would cause division by zero.","points":2,"mcqOptions":null},{"id":"95eef424-a187-41f8-9b5a-172799275de5","cardType":"basic","front":"When can the limit $h\\to 0$ be evaluated in a first-principles calculation?","back":"After the factor of $h$ has cancelled.","points":1,"mcqOptions":null},{"id":"9201a537-5a43-4e06-a286-a50fc0f2ff42","cardType":"cloze","front":"The expansion needed for differentiating $x^2$ is $\\text{[...]}$.","back":"The expansion needed for differentiating $x^2$ is $\\boxed{(x+h)^2=x^2+2xh+h^2}$.","points":null,"mcqOptions":null},{"id":"be1dbff6-d1c0-43fc-91ba-9a48ddb630b9","cardType":"basic","front":"After simplifying and cancelling $h$, what expression remains when differentiating $x^2$ from first principles?","back":"$2x+h$","points":1,"mcqOptions":null},{"id":"c9d781b0-dabd-4b64-a806-1c1e4c997c34","cardType":"basic","front":"What derivative results when $h\\to 0$ in $2x+h$?","back":"$\\frac{d}{dx}(x^2)=2x$","points":1,"mcqOptions":null},{"id":"a34e527c-7907-4914-9780-e9b0d99f3888","cardType":"cloze","front":"The cubic expansion is $\\text{[...]}$.","back":"The cubic expansion is $\\boxed{(x+h)^3=x^3+3x^2h+3xh^2+h^3}$.","points":null,"mcqOptions":null},{"id":"8a514a29-1112-4d22-a44e-aeb30a09af6e","cardType":"basic","front":"After cancelling $h$, what expression remains when differentiating $x^3$ from first principles?","back":"$3x^2+3xh+h^2$","points":1,"mcqOptions":null},{"id":"1565c5ce-a9d3-4928-881a-9602d491d535","cardType":"basic","front":"What derivative results when $h\\to 0$ in $3x^2+3xh+h^2$?","back":"$\\frac{d}{dx}(x^3)=3x^2$","points":1,"mcqOptions":null},{"id":"68f2e5c8-a42e-4920-88fc-d78003fb1d8b","cardType":"cloze","front":"The fourth-power expansion is $\\text{[...]}$.","back":"The fourth-power expansion is $\\boxed{(x+h)^4=x^4+4x^3h+6x^2h^2+4xh^3+h^4}$.","points":null,"mcqOptions":null},{"id":"4946b2d7-0580-47ad-a432-88f114987968","cardType":"basic","front":"After dividing by $h$, what expression remains when differentiating $x^4$ from first principles?","back":"$4x^3+6x^2h+4xh^2+h^3$","points":1,"mcqOptions":