A chemical reaction's equilibrium state is represented by a curve CCC with the equation
2x−xy+y2=28,y≥0 2^x - xy + y^2 = 28, \quad y \ge 0 2x−xy+y2=28,y≥0where xxx and yyy represent the concentrations of two specific reagents in mol dm−3^{-3}−3. A specific experimental state PPP lies on CCC where the concentration x=4x = 4x=4.
Find the concentration yyy at state PPP.
Find dydx\frac{\text{d}y}{\text{d}x}dxdy in terms of xxx and yyy.
The tangent to CCC at PPP intersects the xxx-axis at the point QQQ.
Find the xxx-coordinate of QQQ, giving your answer in the form aln2+bcln2+d\frac{a \ln 2 + b}{c \ln 2 + d}cln2+daln2+b where a,b,ca, b, ca,b,c and ddd are integers to be found.
425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.