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Study SL 5.7—Second derivative and graph behaviour conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

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The derivative of a function f is given by $f^{\prime}(x)=\frac{2 x+2}{x^{2}+2 x+2}$ , for $x \in \mathbb{R}$ .

Show that $f^{\prime \prime}(x)=\frac{-2 x^{2}-4 x}{\left(x^{2}+2 x+2\right)^{2}}$ .

attempt to use the quotient rule

f^{\prime \prime}(x)=\frac{2\left(x^{2}+2 x+2\right)-(2 x+2)(2 x+2)}{\left(x^{2}+2 x+2\right)^{2}}

Note: Award A1 for first term in numerator, A1 for second term in numerator (including the minus sign). If the numerator is correct, but the denominator is incorrect or missing, award M1A1A0.

\begin{aligned} & f^{\prime \prime}(x)=\frac{2 x^{2}+4 x+4-4 x^{2}-8 x-4}{\left(x^{2}+2 x+2\right)^{2}} \text { OR } f^{\prime \prime}(x)=\frac{2 x^{2}+4 x+4-\left(4 x^{2}+8 x+4\right)}{\left(x^{2}+2 x+2\right)^{2}} \\ & f^{\prime \prime}(x)=\frac{-2 x^{2}-4 x}{\left(x^{2}+2 x+2\right)^{2}} \end{aligned}

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IB Maths AA SL 5.7 Second Derivative and Graph Behaviour Topic Practice

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Question 1