The parametric equations of a curve are $$\[ x=t+\ln (t+2), \quad y=(t-1) \mathrm{e}^{-2 t}, \]$$ where $$\(t>-2\)$$. Express $$\(\frac{\mathrm{d} y}{\mathrm{~d} x}\)$$ in terms of $$\(t\)$$, simplifying your answer. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_s21_qp_33 Year：2021 Question No：3(a)
Answer:
State \(\frac{\mathrm{d} x}{\mathrm{~d} t}=1+\frac{1}{t+2}\)
Use product rule
Obtain \(\frac{\mathrm{d} y}{\mathrm{~d} t}=\mathrm{e}^{-2 t}-2(t-1) \mathrm{e}^{-2 t}\)
Use \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d} y}{\mathrm{~d} t} \div
\frac{\mathrm{d} x}{\mathrm{~d} t}\)
Obtain correct answer in any simplified form,
e.g. \(\frac{(3-2 t)(t+2)}{t+3} \mathrm{e}^{-2 t}\)
Use product rule
Obtain \(\frac{\mathrm{d} y}{\mathrm{~d} t}=\mathrm{e}^{-2 t}-2(t-1) \mathrm{e}^{-2 t}\)
Use \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d} y}{\mathrm{~d} t} \div
\frac{\mathrm{d} x}{\mathrm{~d} t}\)
Obtain correct answer in any simplified form,
e.g. \(\frac{(3-2 t)(t+2)}{t+3} \mathrm{e}^{-2 t}\)
Knowledge points:
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
3.4.3 find and use the first derivative of a function which is defined parametrically or implicitly.
Solution:
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