Question

Find the derivative of ${{\sin }^{2}}x$ with respect to x using product rule?

Answer 1

We need to find the derivative of the function ${{\sin }^{2}}x$ . We start to solve the problem by writing ${{\sin }^{2}}x$ as $\sin x\times \sin x$ . Then, we will be using the product rule to find the derivative of the given trigonometric function.

Complete step-by-step solution:
We are given a function and need to find the derivative of it. We solve this question using the product rule of differentiation.
The product rule of differentiation is used to find the derivatives of the product of two or more functions.
Let u, v be the two functions. Then, the derivative of $u\times v$ as per the product rule of differentiation is given as follows,
$\Rightarrow \dfrac{d}{dx}\left( uv \right)=u\dfrac{dv}{dx}+v\dfrac{du}{dx}$
Here,
$\dfrac{d}{dx}\left( uv \right)$ is the derivative of uv with respect to x
$\dfrac{dv}{dx}$ is the derivative of v with respect to x
$\dfrac{du}{dx}$ is the derivative of u with respect to x
According to our question,
We need to find the derivative of ${{\sin }^{2}}x$ .
From mathematics, we know that the value of ${{a}^{2}}$ can be written as $a\times a$
Following the same with the function ${{\sin }^{2}}x$ , we get,
$\Rightarrow {{\sin }^{2}}x=\sin x\times \sin x$
Now, we have to apply the product rule of differentiation to the above equation.
Comparing with the above formula, we have
u = $\sin x$ ;
v = $\sin x$ .
Substituting the values of u, v in the above formula, we get,
$\Rightarrow \dfrac{d}{dx}\left( {{\sin }^{2}}x \right)=\sin x\dfrac{d}{dx}\left( \sin x \right)+\sin x\dfrac{d}{dx}\left( \sin x \right)$
From the formulae of trigonometry, we know that $\dfrac{d}{dx}\left( \sin x \right)=\cos x$
Substituting the same in the above formula, we get,
$\Rightarrow \dfrac{d}{dx}\left( {{\sin }^{2}}x \right)=\sin x\cos x+\sin x\cos x$
Simplifying the above equation, we get,
$\Rightarrow \dfrac{d}{dx}\left( {{\sin }^{2}}x \right)=2\sin x\cos x$
From the formulae of trigonometry, we know that $2\sin x\cos x=\sin 2x$
Substituting the same in the above formula, we get,
$\therefore \dfrac{d}{dx}\left( {{\sin }^{2}}x \right)=\sin 2x$
Hence, the derivative of ${{\sin }^{2}}x$ with respect to x using product rule is sin2x.

Note: We must know that the expression ${{\sin }^{2}}x$ is the same as ${{\left( \sin x \right)}^{2}}$ and so, can be expressed as $\sin x\times \sin x$ . We must know the basic formulae of trigonometry and derivatives to solve the given question quickly. Here we could also use the chain rule of differentiation.