A laser spotlight on a robotic arm traces a path P P\,P on a high-precision sensor wall. The coordinates (x,y)(x, y)(x,y) of the spotlight at time θ \theta\,θ are given by the parametric equations
x=cosec θ,y=cot(θ+π6),π6<θ<π2 x = \text{cosec } \theta, \quad y = \cot \left( \theta + \frac{\pi}{6} \right), \quad \frac{\pi}{6} < \theta < \frac{\pi}{2} x=cosec θ,y=cot(θ+6π),6π<θ<2πFind dydx\displaystyle \frac{dy}{dx}dxdy in terms of θ\thetaθ.
Find an equation for the tangent to the path P P\,P at the point where θ=π3\displaystyle \theta = \frac{\pi}{3}θ=3π. Give your answer in the form y=mx+cy = mx + cy=mx+c, where m m\,m and c c\,c are constants.
Show that all points on the path P P\,P satisfy the equation
y=Ax2−Bx2−1x2−C y = \frac{A x^2 - B\sqrt{x^2 - 1}}{x^2 - C} y=x2−CAx2−Bx2−1where AAA, BBB, and C C\,C are constants to be determined.
717 exam-style questions on Edexcel A Level Maths Differentiation, covering 9.1 Differentiating sin x and cos x, 9.2 Differentiating exponentials and logarithms, 9.3 The Chain Rule, 9.4 The Product Rule, 9.5 The Quotient Rule, 9.6 Differentiating Trigonometric Functions, 9.7 Parametric Differentiation, 9.8 Implicit Differentiation, 9.9 Using Second Derivatives, and 9.10 Rates of Change. Each one has a worked solution and a mark scheme showing where the marks go.