Moderate -0.3 This is a straightforward application of the chain rule followed by the product rule for the second derivative. While it requires careful algebraic manipulation and factorisation, it's a standard textbook exercise with no conceptual difficulty—slightly easier than average due to being purely mechanical differentiation with no problem-solving element.
M1: Attempting to use the chain rule, any form. A1: Any form. For candidates who fully expanded and differentiate a polynomial without factorising, give SC1 for each correct term of second derivative \(56x^6 + 600x^4 + 1800x^2 + 1000\)
Using product rule with \(u = 8x\), \(v = (x^2+5)^3\)
M1
Need not be written explicitly. FT their \(\frac{dy}{dx}\) of correct form
Must be factorised – allow for \((x^2+5)^2(56x^2+40)\)
## Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = 4(x^2+5)^3 \times 2x = 8x(x^2+5)^3$ | M1, A1 | M1: Attempting to use the chain rule, any form. A1: Any form. For candidates who fully expanded and differentiate a polynomial without factorising, give SC1 for each correct term of second derivative $56x^6 + 600x^4 + 1800x^2 + 1000$ |
| Using product rule with $u = 8x$, $v = (x^2+5)^3$ | M1 | Need not be written explicitly. FT their $\frac{dy}{dx}$ of correct form |
| $\frac{d^2y}{dx^2} = 8(x^2+5)^3 + 8x \times 3(x^2+5)^2 \times 2x$ | A1 | Any form |
| $= 8(x^2+5)^3 + 48x^2(x^2+5)^2$ | | |
| $= 8(x^2+5)^2(7x^2+5)$ | A1 [5] | Must be factorised – allow for $(x^2+5)^2(56x^2+40)$ |
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4 Find the second derivative of $\left( x ^ { 2 } + 5 \right) ^ { 4 }$, giving your answer in factorised form.
\hfill \mbox{\textit{OCR MEI Paper 1 2020 Q4 [5]}}