| Directions: (36 - 40) |
| If a relation between x and y is such that y cannot be expressed in terms of x, then y is called an implicit function of x. When a given relation expresses y as an implicit function of x and we want to find \[\frac{dy}{dx}\], then we differentiate every term of the given relation w.r.t x, remembering that a term in y is first differentiated w.r.t. y and then multiplied by \[\frac{dy}{dx}\]. |
| Based on the above information, find the value of \[\frac{dy}{dx}\]in each of the following questions. |
A) \[\frac{\left( 3{{x}^{2}}+2xy+{{y}^{2}} \right)}{{{x}^{2}}+2xy+3{{y}^{2}}}\]
B) \[\frac{-\left( 3{{x}^{2}}+2xy+{{y}^{2}} \right)}{{{x}^{2}}+2xy+3{{y}^{2}}}\]
C) \[\frac{\left( 3{{x}^{2}}+2xy-{{y}^{2}} \right)}{{{x}^{2}}-2xy+3{{y}^{2}}}\]
D) \[\frac{3{{x}^{2}}+xy+{{y}^{2}}}{{{x}^{2}}+xy+3{{y}^{2}}}\]
Correct Answer: B
Solution :
\[{{x}^{3}}+{{x}^{2}}y+x{{y}^{2}}+{{y}^{3}}=81\] \[\Rightarrow 3{{x}^{2}}+{{x}^{2}}\frac{dy}{dx}+2xy+2xy\frac{dy}{dx}+{{y}^{2}}+3{{y}^{2}}\frac{dy}{dx}=0\] \[\Rightarrow \,\left( {{x}^{2}}+2xy+3{{y}^{2}} \right)\frac{dy}{dx}=-{{3}^{2}}-2xy-{{y}^{2}}\] \[\Rightarrow \frac{dy}{dx}=\frac{-\left( 3{{x}^{2}}+2xy+{{y}^{2}} \right)}{{{x}^{2}}+2xy+3{{y}^{2}}}\]
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