The equation of a curve is $$\(y \mathrm{e}^{2 x}-y^{2} \mathrm{e}^{x}=2\)$$. Show that $$\(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{2 y \mathrm{e}^{x}-y^{2}}{2 y-\mathrm{e}^{x}}\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_w21_qp_32 Year：2021 Question No：9(a)
Answer:
State correct derivative of \(y \mathrm{e}^{2 x}\) with respect to \(x\)
State correct derivative of \(y^{2} \mathrm{e}^{x}\) with respect to \(x\)
Equate attempted derivative of the LHS to zero and solve for \(\frac{\mathrm{d} y}{\mathrm{~d} x}\)
Obtain \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{2 y \mathrm{e}^{x}-y^{2}}{2 y-\mathrm{e}^{x}}\)
Alternative method for Question 9(a)
Rearrange as \(y=\frac{2}{\mathrm{e}^{2 x}-y \mathrm{e}^{x}} \Rightarrow \frac{\mathrm{d}}{\mathrm{d} x}\left(\mathrm{e}^{2 x}-y \mathrm{e}^{x}\right)=2 \mathrm{e}^{2 x}-y \mathrm{e}^{x}-\mathrm{e}^{x} \frac{\mathrm{d} y}{\mathrm{~d} x}\)
\(\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{2}{\left(\mathrm{e}^{2 x}-y \mathrm{e}^{x}\right)^{2}} \times\left(2 \mathrm{e}^{2 x}-y \mathrm{e}^{x}-\mathrm{e}^{x} \frac{\mathrm{d} y}{\mathrm{~d} x}\right)\)
Solve for \(\frac{\mathrm{d} y}{\mathrm{~d} x}\)
Obtain \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{2 y \mathrm{e}^{x}-y^{2}}{2 y-\mathrm{e}^{x}}\)
State correct derivative of \(y^{2} \mathrm{e}^{x}\) with respect to \(x\)
Equate attempted derivative of the LHS to zero and solve for \(\frac{\mathrm{d} y}{\mathrm{~d} x}\)
Obtain \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{2 y \mathrm{e}^{x}-y^{2}}{2 y-\mathrm{e}^{x}}\)
Alternative method for Question 9(a)
Rearrange as \(y=\frac{2}{\mathrm{e}^{2 x}-y \mathrm{e}^{x}} \Rightarrow \frac{\mathrm{d}}{\mathrm{d} x}\left(\mathrm{e}^{2 x}-y \mathrm{e}^{x}\right)=2 \mathrm{e}^{2 x}-y \mathrm{e}^{x}-\mathrm{e}^{x} \frac{\mathrm{d} y}{\mathrm{~d} x}\)
\(\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{2}{\left(\mathrm{e}^{2 x}-y \mathrm{e}^{x}\right)^{2}} \times\left(2 \mathrm{e}^{2 x}-y \mathrm{e}^{x}-\mathrm{e}^{x} \frac{\mathrm{d} y}{\mathrm{~d} x}\right)\)
Solve for \(\frac{\mathrm{d} y}{\mathrm{~d} x}\)
Obtain \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{2 y \mathrm{e}^{x}-y^{2}}{2 y-\mathrm{e}^{x}}\)
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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