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.
375 exam-style questions on AQA A Level Maths 1.10 G: Differentiation, covering 1.10.1 The derivative and second derivative, 1.10.2 Differentiating standard functions, 1.10.3 Applications of differentiation, 1.10.4 Product, quotient and chain rules (A-level only), 1.10.5 Implicit and parametric differentiation (A-level only), and 1.10.6 Constructing differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.