Standard +0.3 This is a straightforward application of the product rule followed by a second application to find the second derivative, then substitution of a specific value. While it requires two differentiation steps and involves logarithms, it's a standard textbook exercise with no conceptual challenges—slightly easier than average since the steps are mechanical and the substitution x=e² simplifies the logarithm nicely.
1 Given that \(y = 4 x ^ { 2 } \ln x\), find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = \mathrm { e } ^ { 2 }\).
Attempt use of product rule to find first derivative
M1
producing form \(\ldots \pm \ldots\) where one term involves \(\ln x\) and the other does not
Obtain [first derivative]
A1
or unsimplified equiv
Attempt use of correct product rule to find second derivative
M1
with one term involving \(\ln x\)
Obtain \(8\ln x + 12\)
A1
or unsimplified equiv
Obtain \(28\)
A1
Total
[5]
## Question 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt use of product rule to find first derivative | M1 | producing form $\ldots \pm \ldots$ where one term involves $\ln x$ and the other does not |
| Obtain [first derivative] | A1 | or unsimplified equiv |
| Attempt use of correct product rule to find second derivative | M1 | with one term involving $\ln x$ |
| Obtain $8\ln x + 12$ | A1 | or unsimplified equiv |
| Obtain $28$ | A1 | |
| **Total** | **[5]** | |
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1 Given that $y = 4 x ^ { 2 } \ln x$, find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ when $x = \mathrm { e } ^ { 2 }$.
\hfill \mbox{\textit{OCR C3 2014 Q1 [5]}}