A-Level Mathematics: Differentiation questions

A1: $20x{^3} - 6x$. Accept: Equivalent simplified forms. Notes: Do not award if constant term $7$ is differentiated incorrectly or retained in the derivative.

M1 for differentiating each term correctly: $\frac{d}{dx}(5x{^4})=20x{^3}$, $\frac{d}{dx}(-3x{^2})=-6x$, $\frac{d}{dx}(7)=0$. A1 for combining all terms correctly to give $20x{^3}-6x$. Accept $20x{^3}-6x+0$. Do not award the accuracy mark if a term is omitted or an incorrect power rule is used, e.g. $20x{^4}-6x$ or $20x{^3}-3x$.

M1 for differentiating to obtain $\frac{dy}{dx}=3x{^2}-12x+9$. M1 for solving $3x{^2}-12x+9=0$ to get $x=1$ and $x=3$. A1 for substituting each $x$ value into the original equation correctly. A1 for coordinates $(1,4)$ and $(3,0)$. Accept equivalent ordered pairs in any order. If a candidate finds only the $x$-coordinates, award the two method marks only. If one coordinate is correct and the other is wrong, award ft from the correct $x$-value for the corresponding $y$-value.