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Let y = \sin(\frac{y}{x}) then \frac{\mathrm{d} y}{\mathrm{d} x} equals ?

  • Option 1)

    1 - \cos(\frac{y}{x})

  • Option 2)

    1 + \cos(\frac{y}{x})

  • Option 3)

    \frac{y\cos(\frac{y}{x})}{x^{2} + x\cos(\frac{y}{x})}

  • Option 4)

    \frac{y\cos(\frac{y}{x})}{ x\cos(\frac{y}{x})-x^{2} }

 

Answers (1)

best_answer

As we have learnt,

 

Derivative of implict function -

When  y  is given in any function then we find derivative of function first then find derivative of  y and collect the terms containing  

dy / dx  on left side and find  dy / dx in terms of  x & y

- wherein

ex:

Let \:\:y=siny

\frac{dy}{dx}=cosy\times\frac{dy}{dx}

\frac{dy}{dx}(1-cosy)=0

ex:

Let\:\:y=sin(xy)

\frac{dy}{dx}=cos(xy)\times(1.y+\frac{xdy}{dx})

\frac{dy}{dx}(1-x\:cos(xy))=y\:cos(xy)

\therefore \:\frac{dy}{dx}=\frac{y\:cos(xy)}{1-x\:cos(xy)}

 

 On differentiating both sides we have,

\frac{\mathrm{d} y}{\mathrm{d} x} = \cos\left(\frac{y}{x} \right )\times \left[x\frac{\mathrm{d} y}{\mathrm{d} x} - y \right ] \Rightarrow x^{2}\frac{\mathrm{d} y}{\mathrm{d} x} = x\cos\left (\frac{y}{x} \right ) - y\cos\left(\frac{y}{x} \right )

\frac{\mathrm{d} y}{\mathrm{d} x}\left(x\cos\left(\frac{y}{x} -x^2 \right ) \right ) = y\cos\left(\frac{y}{x} \right )\Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x} = \frac{y\cos\left(\frac{y}{x} \right )}{x\cos\left(\frac{y}{x} \right ) - x^{2}}

 


Option 1)

1 - \cos(\frac{y}{x})

Option 2)

1 + \cos(\frac{y}{x})

Option 3)

\frac{y\cos(\frac{y}{x})}{x^{2} + x\cos(\frac{y}{x})}

Option 4)

\frac{y\cos(\frac{y}{x})}{ x\cos(\frac{y}{x})-x^{2} }

Posted by

Himanshu

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