A curve has the equation
y=ax2y = a^{x^2}y=ax2
where a a\,a is a constant greater than 1.
Show that dydx=2xax2lna\dfrac{dy}{dx} = 2xa^{x^2}\ln adxdy=2xax2lna.
The tangent to the curve at the point (1,a)(1, a)(1,a) passes through the point (12,0)\displaystyle \left(\frac{1}{2}, 0\right)(21,0). Find the value of aaa, giving your answer in exact form.
By considering d2ydx2\dfrac{d^2y}{dx^2}dx2d2y, show that the curve is convex for all values of xxx.
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.