A sensor tracking the movement of a robotic probe in a research lab models the horizontal displacement xxx and vertical displacement yyy at time ttt using the parametric equations
x=7×6−t+3 x = 7 \times 6^{-t} + 3 x=7×6−t+3 y=4×6t−5 y = 4 \times 6^{t} - 5 y=4×6t−5Show that dydx=−47×62t\dfrac{dy}{dx} = -\dfrac{4}{7} \times 6^{2t}dxdy=−74×62t.
Find the Cartesian equation of the curve in the form xy+ax+by=cxy + ax + by = cxy+ax+by=c, where aaa, bbb and ccc are integers.
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.