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.
$\Rightarrow {\displaystyle \frac{du}{dx}}={\displaystyle \frac{d{\sin }^{2}x}{dx}}$
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,
$\Rightarrow {\displaystyle \frac{du}{dx}}=2\sin x.\cos x=\sin 2x\{\because \sin 2x=2\sin x.\cos x\}$…………….. $\left(1\right)$
Now, we will differentiate v with respect to x.
$\Rightarrow {\displaystyle \frac{dv}{dx}}={\displaystyle \frac{d{\cos }^{2}x}{dx}}$
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,
$\Rightarrow {\displaystyle \frac{dv}{dx}}=2\cos x\times (-\sin x)=-\sin 2x\{\because \sin 2x=2\sin x.\cos x\}$ …………….. $\left(2\right)$
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,
$\Rightarrow {\displaystyle \frac{du}{dv}}={\displaystyle \frac{\sin 2x}{-\sin 2x}}$
On dividing numerator by denominator, we get
$\Rightarrow {\displaystyle \frac{du}{dv}}=-1$
We have found ${\displaystyle \frac{du}{dv}}$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 $f^{\prime }(g(x)).g^{\prime }(x)$