al:\n\n$$\\begin{aligned} \\frac{dy}{dx} &= \\frac{d}{dx}(x^6) + \\frac{d}{dx}(15x^4) + \\frac{d}{dx}(75x^2) + \\frac{d}{dx}(125) \\\\ &= 6x^5 + 15(4x^3) + 75(2x) + 0 \\\\ &= 6x^5 + 60x^3 + 150x \\end{aligned}$$"},{"type":"text","order":3,"title":"Factorise the derivative","content":"To match the standard factored form of the derivative (which is typically expected in exams), we can factor out the common term $6x$:\n\n$$\\begin{aligned} \\frac{dy}{dx} &= 6x(x^4 + 10x^2 + 25) \\end{aligned}$$\n\nNotice that the quadratic in the parentheses is a perfect square trinomial:\n\n$$x^4 + 10x^2 + 25 = (x^2 + 5)^2$$\n\nSubstituting this back gives our final answer:\n\n$$\\frac{dy}{dx} = 6x(x^2 + 5)^2$$"}]}],"generatedAt":"2026-06-09T08:10:21.305Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"324c01fc-3c23-4838-a251-2b61cef38926","revisionId":"41c0e52d-87b8-4b90-870a-97d6331737a7","questionNumber":null,"questionType":"LA","specification":"The point $P$ lies on the curve with equation $y = \\sec x + 2\\sin 2x$ with $x$-coordinate $\\frac{\\pi}{6}$","difficulty":2,"difficultyBand":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"79c5cd78-9616-418e-ab25-64d0d3c2f26c","label":"a","content":"Find an equation to the tangent to the curve at the point $P$.","markscheme":"- Differentiates to find $\\frac{dy}{dx} = \\sec x \\tan x + 4\\cos 2x$ M1\n- Substitutes $x = \\frac{\\pi}{6}$ to find gradient $m = (\\frac{2}{\\sqrt{3}})(\\frac{1}{\\sqrt{3}}) + 4(\\frac{1}{2}) = \\frac{2}{3} + 2 = \\frac{8}{3}$ M1\n- Finds $y$-coordinate: $y = \\sec(\\frac{\\pi}{6}) + 2\\sin(\\frac{\\pi}{3}) = \\frac{2}{\\sqrt{3}} + \\sqrt{3} = \\frac{5\\sqrt{3}}{3}$ B1\n- Uses $y - y_1 = m(x - x_1)$ with their values: $y - \\frac{5\\sqrt{3}}{3} = \\frac{8}{3}(x - \\frac{\\pi}{6})$ M1\n- Correct method for simplifying or finding $c$ M1\n- Final equation: $y = \\frac{8}{3}x + \\frac{5\\sqrt{3}}{3} - \\frac{4\\pi}{9}$ or equivalent A1","marks":6,"order":0,"markingRubric":{"totalMarks":6,"points":[{"id":"p1","marks":1,"description":"Differentiates the given function to find dy/dx.","code":"M"},{"id":"p2","marks":1,"description":"Calculates the numerical gradient m at x = pi/6.","code":"M"},{"id":"p3","marks":1,"description":"Calculates the y-coordinate at the point P.","code":"B"},{"id":"p4","marks":1,"description":"Applies the point-slope form y - y1 = m(x - x1).","code":"M"},{"id":"p5","marks":1,"description":"Uses a correct method to solve for c or rearrange the equation.","code":"M"},{"id":"p6","marks":1,"description":"Provides the correct final tangent equation.","code":"A"}],"finalAnswers":["y = \\frac{8}{3}x + \\frac{5\\sqrt{3}}{3} - \\frac{4\\pi}{9}","y = \\frac{8}{3}x + \\frac{15\\sqrt{3} - 4\\pi}{9}"],"generatedAt":"2026-05-06T14:04:20.685Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"analytical","label":"Analytical Differentiation","steps":[{"type":"text","order":0,"title":"Identify the goal","content":"To find the equation of the tangent line to the curve $y = \\sec x + 2\\sin 2x$ at $x = \\frac{\\pi}{6}$, we will use the point-slope form of a linear equation:\\n\\n$$y - y_1 = m(x - x_1)$$\\n\\nTo do this, we need to find:\\n1. The $y$-coordinate of the point of contact ($y_1$)\\n2. The gradient of the tangent line at that point ($m$), which we obtain by differentiating $y$ and substituting $x = \\frac{\\pi}{6}$."},{"type":"text","order":1,"title":"Find the y-coordinate","content":"Substitute $x = \\frac{\\pi}{6}$ into the original equation to find the $y$-coordinate of $P$:\\n\\n$$y = \\sec\\left(\\frac{\\pi}{6}\\right) + 2\\sin\\left(2 \\cdot \\frac{\\pi}{6}\\right)$$\\n\\nRecall the exact trigonometric values:\\n* $\\sec\\left(\\frac{\\pi}{6}\\right) = \\frac{1}{\\cos\\left(\\frac{\\pi}{6}\\right)} = \\frac{1}{\\frac{\\sqrt{3}}{2}} = \\frac{2}{\\sqrt{3}} = \\frac{2\\sqrt{3}}{3}$\\n* $\\sin\\left(\\frac{\\pi}{3}\\right) = \\frac{\\sqrt{3}}{2}$\\n\\nNow substitut

9.3 The Chain Rule Questions

9.3 The Chain Rule Questions

9.3 The Chain Rule Questions

9.3 The Chain Rule Questions