Differentiation from first principlesEdexcel A-Level Maths: Mind map

Exam questions on Differentiation from first principles

1. The point $P(3,9)$ lies on the curve $y=x^{2}$. The point $Q$ on the same curve has $x$-coordinate $3+h$, where $h\neq0$.
   Find the equation of the tangent to the curve at $P$.2 marks
2. Let $f(x)=x^{3}$.
   Given that $f'(x)=3x^{2}$, find $f''(x)$, and hence find the rate of change of the gradient of the curve $y=f(x)$ when $x=2$.2 marks
3. A curve has equation $y=x^{2}$.
   Prove, from first principles, that $\frac{\mathrm{d}y}{\mathrm{d}x}=2x$.3 marks