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The value of $y^{\prime} / x^{\prime}$ in terms of the angle 0 is given by
a) $\tan \theta$
b) $\sec \theta$
c) $\cot \theta$
d) $\operatorname{cosec} \theta$
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(a)

Explanation:

The value of derivative of a function $f(x)$ is given as $f^{\prime}(x)=y^{\prime} / x^{\prime}$. In terms of theta tangent is the ration of opposite side to adjacent side hence $y^{\prime} / x^{\prime}=\tan \theta$.

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The value of $y^{\prime} / x^{\prime}$ is the derivative of $y$ with respect to $x$, often denoted as $d y / d x$. In the context of trigonometry and polar coordinates, this ratio is often associated with the tangent of an angle.
If you're referring to the angle $\theta$ (theta), then the ratio $y^{\prime} / x^{\prime}$ is equivalent to the tangent of the angle $\theta$, denoted as $\tan (\theta)$.

This is because in a right triangle, the tangent of an angle is defined as the ratio of the side opposite the angle (often denoted as $y$ ) to the side adjacent to the angle (often denoted as $x$ ).
So, $y^{\prime} / x^{\prime}=\tan (\theta)$
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