A curve has equation
y2=ye3x+2x y^2 = y e^{3x} + 2x y2=ye3x+2xShow that
dydx=3ye3x+22y−e3x \frac{dy}{dx} = \frac{3y e^{3x} + 2}{2y - e^{3x}} dxdy=2y−e3x3ye3x+2The curve crosses the yyy-axis at the origin (0,0)(0, 0)(0,0) and at a second point PPP. The tangent to the curve at the origin and the tangent to the curve at PPP meet at the point RRR. Find the coordinates of RRR.
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.