Question

If $x = {\cos ^2}\theta$ and $y = \cot \theta$, then find $\dfrac{{dy}}{{dx}}$at $\theta = \dfrac{\pi }{4}$

Answer 1

Here, we will use the concept of differentiation to find the value of $\dfrac{{dy}}{{dx}}$. We will first find the value of the $dx$ and then we will find the value of the $dy$ separately. Then by dividing their values we will get the value of $\dfrac{{dy}}{{dx}}$. Further, we will substitute the value of $\theta = \dfrac{\pi }{4}$ to get the final answer.

Complete step-by-step answer:
We will first find the value of $dx$ by differentiating the equation ,$x = {\cos ^2}\theta$.
We know that the differentiation of $\cos \theta$ is $- \sin \theta$.
Differentiating the equation $x = {\cos ^2}\theta$, we get
$dx = - 2\cos \theta \sin \theta \,d\theta$……………….$\left( 1 \right)$
Now we will find the value of $dy$ by differentiating the equation $y = \cot \theta$.
We know that the differentiation of $\cot \theta$ is $- \cos e{c^2}\theta$. So, we get
$dy = - \cos e{c^2}\theta \,d\theta$……………….$\left( 2 \right)$
Now by dividing the equation $\left( 2 \right)$ by $\left( 1 \right)$, we get
$\dfrac{{dy}}{{dx}} = \dfrac{{ - \cos e{c^2}\theta \,d\theta }}{{ - 2\cos \theta \sin \theta \,d\theta }}$
We know that ${\rm{cosec}}\theta$ is the reciprocal of the $\sin \theta$. Therefore, we get
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{2{{\sin }^2}\theta \cos \theta \sin \theta }}$
Now we will find the value of the $\dfrac{{dy}}{{dx}}$ at $\theta = \dfrac{\pi }{4}$.
We know that the value of $\sin \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}$ and $\cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}$.
Substituting $\sin \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}$ and $\cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}$ in the equation, we get
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{2 \times {{\left( {\dfrac{1}{{\sqrt 2 }}} \right)}^2} \times \dfrac{1}{{\sqrt 2 }} \times \dfrac{1}{{\sqrt 2 }}}}$
Simplifying the equation, we get
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{2 \times \dfrac{1}{2} \times \dfrac{1}{2}}} = 2$
Hence, $\dfrac{{dy}}{{dx}}$ at $\theta = \dfrac{\pi }{4}$ is 2.

Note: Here, we need to know the basic differentiation of the trigonometric function in order to solve questions. We have used differentiation by parts to find the value of $dx$. Differentiation is a method by which we can measure per unit of a function in the given independent variable.