cos \\theta} = \\tan \\theta$ to write the final simplified answer (earning the final mark A1)."}]}],"generatedAt":"2026-06-09T02:01:03.981Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"ef7406ef-b962-485d-967b-263dbf039539","revisionId":"319bfec9-8d97-465d-8a74-bfb14d61853e","questionNumber":null,"questionType":"LA","specification":"A curve has the equation $x^2 + 2xy - y^2 = 1$:","difficulty":1,"difficultyBand":"medium","totalMarks":8,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"5903cf30-ffbf-46a9-a97a-af20fcd22227","label":"a","content":"Find an expression for $\\frac{dy}{dx}$","markscheme":"- Differentiates $x^2$ to $2x$ and $y^2$ to $2y \\frac{dy}{dx}$ M1\n- Applies product rule to $2xy$ to get $2y + 2x \\frac{dy}{dx}$ M1\n- Sets the derivative of the constant 1 to 0 B1\n- Rearranges to factorise $\\frac{dy}{dx}$ M1\n- Correct final expression $\\frac{dy}{dx} = \\frac{x + y}{y - x}$ or equivalent A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Differentiate $x^2$ and $y^2$ terms correctly with respect to $x$.","code":"M"},{"id":"p2","marks":1,"description":"Correct application of product rule to the $2xy$ term.","code":"M"},{"id":"p3","marks":1,"description":"Derivative of the constant on the RHS is set to zero.","code":"B"},{"id":"p4","marks":1,"description":"Isolate terms containing $dy/dx$ and factorise correctly.","code":"M"},{"id":"p5","marks":1,"description":"Final simplified expression for $dy/dx$.","code":"A"}],"finalAnswers":["\\frac{dy}{dx} = \\frac{x + y}{y - x}","\\frac{dy}{dx} = \\frac{-x - y}{x - y}"],"generatedAt":"2026-05-06T14:04:17.008Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"implicit_differentiation_x","label":"Implicit Differentiation with respect to x","steps":[{"type":"text","order":0,"title":"Identify the method","content":"To find $\\frac{dy}{dx}$ for the equation \n\n$$x^2 + 2xy - y^2 = 1$$\n\nwe will perform **implicit differentiation with respect to $x$**. This means we will differentiate every term on both sides of the equation with respect to $x$, applying the **product rule** and the **chain rule** where necessary.","highlights":[{"text":"x^2 + 2xy - y^2 = 1","location":"specification"}]},{"type":"text","order":1,"title":"Differentiate simple terms","content":"Let's first differentiate the simpler terms on both sides of the equation with respect to $x$:\n- The derivative of $x^2$ is $2x$.\n- Using the chain rule, the derivative of $y^2$ with respect to $x$ is $$2y \\frac{dy}{dx}$$\n- The derivative of the constant on the right side, $1$, is $0$.\n\nThese individual steps will earn you the first **M1** and **B1** marks."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\frac{d}{dx}(\\htmlData{anchor=u-part}{\\color{@sky}{2x}} \\cdot \\htmlData{anchor=v-part}{\\color{@purple}{y}})$$"},{"latex":"$$$= \\htmlData{anchor=u-diff}{\\color{@sky}{2}}\\htmlData{anchor=v-keep}{\\color{@purple}{y}} + \\htmlData{anchor=u-keep}{\\color{@sky}{2x}}\\htmlData{anchor=v-diff}{\\color{@purple}{\\frac{dy}{dx}}}$$"}],"highlights":[{"color":"sky","style":"box","anchor":"u-part"},{"color":"purple","style":"box","anchor":"v-part"}],"connections":[{"to":"u-diff","from":"u-part","color":"sky","label":"differentiate","style":"solid"},{"to":"v-diff","from":"v-part","color":"purple","label":"differentiate","style":"solid"}]},"order":2,"title":"Apply the product rule","description":"Next, we apply the product rule to the term $2xy$ by letting $u = 2x$ and $v = y$."},{"type":"text","order":3,"title":"Assemble the equation","content":"Now we combine all of our derivatives into a single equation:\n\n$$2x + \\left(2y + 2x\\frac{dy}{dx}\\right) - 2y\\frac{dy}{dx} = 0$$\n\nWhich simplifies to:\n\n$$2x + 2y + 2x\\frac{dy}{dx} - 2y\\frac{dy}{dx} = 0$$"},{"type":"text","order":4,"title":"Factorise and isolate","content":"To solve for $\\frac{dy}{dx}$, we group the terms containing $

Differentiation

Question 28

Differentiation Questions

Differentiation

Question 28

Differentiation Questions