| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| Topic | Differentiation from First Principles |
| Type | First principles: reciprocal function |
| Difficulty | Standard +0.3 This is a guided first-principles differentiation question where part (i) does the algebraic heavy lifting for you. Students just need to apply the definition of derivative and use the given identity. While first principles questions require careful algebra, the scaffolding makes this easier than average. |
| Spec | 1.07g Differentiation from first principles: for small positive integer powers of x |
**(i)** Attempt to form LCM and cross multiply — M1
Attempt to expand bracket and simplify — M1
Obtain given answer — A1 **[3]**
**(ii)** State $\dfrac{1}{h}\left(\dfrac{1}{(x+h)^2} - \dfrac{1}{x^2}\right)$ or equivalent form — M1
Attempt to substitute the AG and obtain $\dfrac{-2x - h}{x^2(x+h)^2}$ — M1
Obtain $-2x^{-3}$ with full and accurate notation in the proof throughout — A1 **[3]** **[6]**12 (i) Prove the identity $\frac { 1 } { ( x + h ) ^ { 2 } } - \frac { 1 } { x ^ { 2 } } \equiv \frac { - 2 h x - h ^ { 2 } } { x ^ { 2 } ( x + h ) ^ { 2 } }$.\\
(ii) Given that $\mathrm { f } ( x ) = x ^ { - 2 }$, use differentiation from first principles to find an expression for $\mathrm { f } ^ { \prime } ( x )$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2013 Q12 [6]}}