The trajectory of a particle in a high-energy magnetic field is described by the implicit curve C C\,C defined by the equation
x2siny+y2cosx=K x^2 \sin y + y^2 \cos x = K x2siny+y2cosx=Kwhere K K\,K is a constant. The particle is observed to pass through the point P(π,π2)\displaystyle P\left(\pi, \frac{\pi}{2}\right)P(π,2π).
Show that K=3π24\displaystyle K = \frac{3\pi^2}{4}K=43π2.
Show that dydx=y2sinx−2xsinyx2cosy+2ycosx\displaystyle \frac{dy}{dx} = \frac{y^2 \sin x - 2x \sin y}{x^2 \cos y + 2y \cos x}dxdy=x2cosy+2ycosxy2sinx−2xsiny.
Hence, determine the numerical gradient of the trajectory at point PPP.
The tangent to the trajectory at P P\,P intersects the xxx-axis at the point QQQ. Find the exact xxx-coordinate of QQQ.
333 exam-style questions on WJEC A Level Maths 3.6 Differentiation (A-level only), covering 3.6.1 Differentiation (A-level only), 3.6.2 Differentiation (A-level only), 3.6.3 Differentiation (A-level only), 3.6.4 Differentiation (A-level only), 3.6.5 Differentiation (A-level only), 3.6.6 Differentiation (A-level only), 3.6.7 Differentiation (A-level only), and 3.6 Differentiation (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.