Correct Answer - Option 4 : -x/y
Concept:
Steps for derivatives of functions expressed in the parametric form:
- First of all, we write the given functions u and v in terms of the parameter x.
- Using differentiation find out du/dx and dv/dx.
- Then by using the formula used for solving functions in parametric form i.e. \(\frac{{{\rm{du}}}}{{{\rm{dv}}}} = \frac{{\left( {\frac{{{\rm{du}}}}{{{\rm{dx}}}}} \right)}}{{\left( {\frac{{{\rm{dv}}}}{{{\rm{dx}}}}} \right)}}{\rm{\;}}\)
- 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.