The curve C C\,C has the equation
x=2tan2yx = 2\tan 2yx=2tan2y, where −π4<y<π4\displaystyle -\frac{\pi}{4} < y < \frac{\pi}{4}−4π<y<4π
You may use the result ddu(tanku)=ksec2ku\dfrac{d}{du}(\tan ku) = k\sec^2 kudud(tanku)=ksec2ku.
Show that, for all points (x,y)(x, y)(x,y) lying on CCC,
dydx=ax2+b\displaystyle \frac{dy}{dx} = \frac{a}{x^2 + b}dxdy=x2+ba
where a a\,a and b b\,b are constants 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.