A micro-robotic probe moves along a path C C\,C within a high-precision magnetic field. The relationship between its horizontal displacement x x\,x and its vertical displacement y y\,y is defined by the equation
x=3sec24y,x>3,0<y<π8 x = 3\sec^2 4y, \quad x > 3, \quad 0 < y < \frac{\pi}{8} x=3sec24y,x>3,0<y<8πFind an expression for dxdy\dfrac{dx}{dy}dydx in terms of yyy.
Hence show that
dydx=pqxx−3 \dfrac{dy}{dx} = \frac{p}{qx\sqrt{x-3}} dxdy=qxx−3pwhere p p\,p is irrational and q q\,q is an integer, stating the values of p p\,p and qqq.
Find the equation of the normal to the probe's path C C\,C at the point where y=π12\displaystyle y = \frac{\pi}{12}y=12π, giving your answer in the form y=mx+cy = mx + cy=mx+c where m m\,m and c c\,c are exact constants.
Practise Edexcel A Level Maths Differentiation with exam-style questions for A Level Maths. 327 questions covering 12.1 Gradients of Curves, 12.2 Differentiation from first principles, 12.3 Differentiating x^n, 12.4 Differentiating Quadratics, 12.5 Differentiating functions with two or more terms, 12.6 Gradients, Tangents and Normals, 12.7 Increasing and Decreasing Functions, 12.8 Second Order Derivatives, 12.9 Stationary Points, 12.10 Sketching Gradient Functions, and 12.11 Modelling with Differentiation, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
