Question

The derivative of ${{\sin }^{2}}x$ with respect to ${{\cos }^{2}}x$ is
A) ${{\tan }^{2}}x$
B) $\tan x$
C) $-\tan x$
D) None of these

Answer 1

Hint:
Here we have to find the derivative of ${{\sin }^{2}}x$ with respect to ${{\cos }^{2}}x$. For that, we will find the differentiation of ${{\sin }^{2}}x$ with respect to x using the chain rule of differentiation by first differentiating ${{\sin }^{2}}x$ and then multiply it with the differentiation of$\sin x$. Similarly, we will find the differentiation of ${{\cos }^{2}}x$ with respect to x using the chain rule of differentiation by first differentiating ${{\cos }^{2}}x$ and multiply it with the differentiation of $\cos x$. Then we will divide the value of the derivative of ${{\sin }^{2}}x$ obtained by the value of the derivative of ${{\cos }^{2}}x$. From there, we will get the result of the derivative of ${{\sin }^{2}}x$ with respect to ${{\cos }^{2}}x$.

Complete step by step solution:
Let ${{\sin }^{2}}x=u$and ${{\cos }^{2}}x=v$
Now, we will first differentiate u with respect to x.
$\Rightarrow \dfrac{du}{dx}=\dfrac{d{{\sin }^{2}}x}{dx}$
Here, we will use the chain rule of differentiation. We will first find the derivative of ${{\sin }^{2}}x$ and then multiply it with the differentiation of$\sin x$
Therefore,
$\Rightarrow \dfrac{du}{dx}=2\sin x.\cos x=\sin 2x\text{ }\left\{ \because \sin 2x=2\sin x.\cos x \right\}$…………….. $\left( 1 \right)$
Now, we will differentiate v with respect to x.
$\Rightarrow \dfrac{dv}{dx}=\dfrac{d{{\cos }^{2}}x}{dx}$
Here, we will use the chain rule of differentiation. We will first find the derivative of ${{\cos }^{2}}x$ and then multiply it with the differentiation of$\cos x$
Therefore,
$\Rightarrow \dfrac{dv}{dx}=2\cos x\times \left( -\sin x \right)=-\sin 2x\text{ }\left\{ \because \sin 2x=2\sin x.\cos x \right\}$ …………….. $\left( 2 \right)$
According to the question, we have to find the derivative of ${{\sin }^{2}}x$ with respect to ${{\cos }^{2}}x$
For that, we will divide equation $\left( 1 \right)$by equation $\left( 2 \right)$
Therefore,
$\Rightarrow \dfrac{du}{dv}=\dfrac{\sin 2x}{-\sin 2x}$
On dividing numerator by denominator, we get
$\Rightarrow \dfrac{du}{dv}=-1$
We have found $\dfrac{du}{dv}$which is equal to derivative of ${{\sin }^{2}}x$ with respect to ${{\cos }^{2}}x$
Therefore, the derivative of ${{\sin }^{2}}x$ with respect to ${{\cos }^{2}}x$is -1.

Hence, the correct option is D.

Note:
We need to know the following terms as we have used them in this solution.
Differentiation is defined as a process in which we find the function which gives the output of rate of change of one variable with respect to another variable.
Differentiation by chain rule: In this method, the differentiation of function $f(g(x))$ is equal to $f'(g(x)).g'(x)$