The diagram shows the curve with parametric equations $$\[ x=\ln (2 t+3), \quad y=\frac{2 t-3}{2 t+3} . \]$$ The curve crosses the $$\(y\)$$-axis at the point $$\(A\)$$ and the $$\(x\)$$-axis at the point $$\(B\)$$. Show that $$\(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6}{2 t+3}\)$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................
Exam No：9709_w21_qp_23 Year：2021 Question No：5(a)
Answer:
Obtain \(\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{2}{2 t+3}\)
Use quotient rule, or equivalent, to find \(\frac{\mathrm{d} y}{\mathrm{~d} t}\)
Obtain \(\frac{\mathrm{d} y}{\mathrm{~d} t}=\frac{2(2 t+3)-2(2 t-3)}{(2 t+3)^{2}}\)
Divide to confirm \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6}{2 t+3}\)
Use quotient rule, or equivalent, to find \(\frac{\mathrm{d} y}{\mathrm{~d} t}\)
Obtain \(\frac{\mathrm{d} y}{\mathrm{~d} t}=\frac{2(2 t+3)-2(2 t-3)}{(2 t+3)^{2}}\)
Divide to confirm \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6}{2 t+3}\)
Knowledge points:
2.4.1 use the derivatives of ,ln x,sin x,cos x,tan x, together with constant multiples, sums, differences and composites
2.4.2 differentiate products and quotients
2.4.3 find and use the first derivative of a function which is defined parametrically or implicitly.
Solution:
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