| Exam Board | OCR |
|---|---|
| Module | PURE |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Differentiate exponential functions |
| Difficulty | Easy -1.2 This is a straightforward differentiation and integration question using basic power rule only. Despite the title mentioning transcendental functions, the actual question involves only polynomial terms (x and x^{-3}), requiring simple application of standard rules with no problem-solving or conceptual challenge. Easier than average A-level content. |
| Spec | 1.07e Second derivative: as rate of change of gradient1.07i Differentiate x^n: for rational n and sums1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(3x\) correctly differentiated to \(3\) | B1 (1.1) | |
| \(kx^{-4}\) oe | M1 (1.1a) | |
| \(15x^{-4}\) or \(\frac{15}{x^4}\) oe | A1 (1.1) | ISW e.g. \(5 \times 3x^{-4} = \frac{5}{3x^4}\) M1A1 |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Zero term stated or implied by absence | B1f (1.1a) | Or ft their 1st term in (a) |
| Other term, ft their 2nd term in (a) if includes negative index | B1f (1.1) | ISW |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{3x^2}{2}\) | B1 (1.1) | |
| \(kx^{-2}\) oe | M1 (1.1a) | |
| All correct and \(+ c\) | A1 (1.1) | ISW |
| [3] |
## Question 1:
*Note: In all three parts: Allow unsimplified answers. ISW for incorrect simplification. Ignore incorrect dx and/or ∫*
### Part (a)
$3 + 15x^{-4}$ or $3 + \frac{15}{x^4}$ oe
| Answer | Mark | Guidance |
|--------|------|----------|
| $3x$ correctly differentiated to $3$ | B1 (1.1) | |
| $kx^{-4}$ oe | M1 (1.1a) | |
| $15x^{-4}$ or $\frac{15}{x^4}$ oe | A1 (1.1) | ISW e.g. $5 \times 3x^{-4} = \frac{5}{3x^4}$ M1A1 |
| **[3]** | | |
### Part (b)
$-60x^{-5}$ or $-\frac{60}{x^5}$ oe
| Answer | Mark | Guidance |
|--------|------|----------|
| Zero term stated or implied by absence | B1f (1.1a) | Or ft their 1st term in (a) |
| Other term, ft their 2nd term in (a) if includes negative index | B1f (1.1) | ISW |
| **[2]** | | |
### Part (c)
$\frac{3x^2}{2} + \frac{5}{2}x^{-2} + c$ or $\frac{3x^2}{2} + \frac{5}{2x^2} + c$ oe
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{3x^2}{2}$ | B1 (1.1) | |
| $kx^{-2}$ oe | M1 (1.1a) | |
| All correct and $+ c$ | A1 (1.1) | ISW |
| **[3]** | | |
---1 It is given that $\mathrm { f } ( x ) = 3 x - \frac { 5 } { x ^ { 3 } }$.\\
Find
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { f } ^ { \prime } ( x )$,
\item $\mathrm { f } ^ { \prime \prime } ( x )$,
\item $\int \mathrm { f } ( x ) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{OCR PURE Q1 [8]}}