$$\( \)$$ Let $$\(\mathrm{f}(x)=\frac{\mathrm{e}^{2 x}+1}{\mathrm{e}^{2 x}-1}\)$$, for $$\(x>0\)$$ Find $$\(\mathrm{f}^{\prime}(x)\)$$. Hence find the exact value of $$\(x\)$$ for which $$\(\mathrm{f}^{\prime}(x)=-8\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_m21_qp_32 Year：2021 Question No：9(c)
Answer:
Use quotient rule
Obtain correct derivative in any form
Equate derivative to \(-8\) and obtain a quadratic in \(\mathrm{e}^{2 x}\)
Obtain \(2\left(\mathrm{e}^{2 x}\right)^{2}-5 \mathrm{e}^{2 x}+2=0\)
Solve a 3-term quadratic in \(\mathrm{e}^{2 x}\) for \(x\)
Obtain answer \(x=\frac{1}{2} \ln 2\), or exact equivalent, only
Alternative method for question \(9(c)\)
Use quotient rule
Obtain correct derivative in any form
Equate derivative to \(-8\), take square roots and obtain a quadratic in \(\mathrm{e}^{x}\)
Obtain \(\sqrt{2} \mathrm{e}^{2 x}-\mathrm{e}^{x}-\sqrt{2}=0\)
Solve a 3-term quadratic in \(\mathrm{e}^{x}\) for \(x\)
Obtain answer \(x=\frac{1}{2} \ln 2\), or exact equivalent, only
Obtain correct derivative in any form
Equate derivative to \(-8\) and obtain a quadratic in \(\mathrm{e}^{2 x}\)
Obtain \(2\left(\mathrm{e}^{2 x}\right)^{2}-5 \mathrm{e}^{2 x}+2=0\)
Solve a 3-term quadratic in \(\mathrm{e}^{2 x}\) for \(x\)
Obtain answer \(x=\frac{1}{2} \ln 2\), or exact equivalent, only
Alternative method for question \(9(c)\)
Use quotient rule
Obtain correct derivative in any form
Equate derivative to \(-8\), take square roots and obtain a quadratic in \(\mathrm{e}^{x}\)
Obtain \(\sqrt{2} \mathrm{e}^{2 x}-\mathrm{e}^{x}-\sqrt{2}=0\)
Solve a 3-term quadratic in \(\mathrm{e}^{x}\) for \(x\)
Obtain answer \(x=\frac{1}{2} \ln 2\), or exact equivalent, only
Knowledge points:
3.2.3 use logarithms to solve equations and inequalities in which the unknown appears in indices
3.4.1 use the derivatives of together with constant multiples, sums, differences and composites (Derivatives of are not required.)
3.4.2 differentiate products and quotients
Solution:
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