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If y = cos t and x = sin t, then what is \(\rm \dfrac{dy}{dx}\) equal to?
1. xy
2. x/y
3. -y/x
4. -x/y
5. None of these

1 Answer

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Best answer
Correct Answer - Option 4 : -x/y

Concept:

Steps for derivatives of functions expressed in the parametric form:

  1. First of all, we write the given functions u and v in terms of the parameter x.
  2. Using differentiation find out du/dx and dv/dx.
  3. Then by using the formula used for solving functions in parametric form i.e. \(\frac{{{\rm{du}}}}{{{\rm{dv}}}} = \frac{{\left( {\frac{{{\rm{du}}}}{{{\rm{dx}}}}} \right)}}{{\left( {\frac{{{\rm{dv}}}}{{{\rm{dx}}}}} \right)}}{\rm{\;}}\)
  4.  Lastly substituting the values of du/dx and dv/dx and simplify to obtain the result.


Calculation:

Here, y = cos t 

Differntiating w.r.t. t, we get 

\(\rm \dfrac{dy}{dt}\) = - sin t         .... (1)

And x = sin t

Differntiating w.r.t. t, we get 

\(\rm \dfrac{dx}{dt}\) = cos t          .... (2)

Now divide (1) by (2), we get

\(\rm \dfrac{dy}{dx}\) = - sin t/ cos t

= -x/y

Hence, option (4) is correct.

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