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Find the second and third derivative of a function\[y=\sin x\].

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Hint: You can use the first principle method to find derivatives of \[\sin x\] and
 \[\cos x\]or directly use the differentiation of \[\sin x\]and \[\cos x\]wherever required.


We have the given function as –
\[y=\sin x-(1)\]
We know that differentiation of any function can be calculated with the help of first principle method of differentiation as stated below: -
If we have a function \[f\left( x \right)\] which is continuous and differentiable for any real number then differentiation of it at any point \[c\] can be stated as:
\[{{f}^{'}}(c)=\underset{x\to c}{\mathop{\lim }}\,\dfrac{f(x)-f(c)}{x-c}-(2)\]
Applying LHL (left hand limit) and RHL (right hand limit) to equation (2) as,
For RHL:
\[\underset{h\to 0}{\mathop{\lim }}\,\dfrac{f(c+h)-f(c)}{h}\]
And LHL can be written as,
\[\underset{h\to 0}{\mathop{\lim }}\,\dfrac{f(c-h)-f(c)}{-h}\]
And we can verify that values got from LHL and RHL should be equal.
Hence, applying first principle method of differentiation we can find derivative of,
\[y=\sin x\]as \[\cos x\].
Hence, \[\dfrac{dy}{dx}=\dfrac{d}{dx}(\sin x)=\cos x-(3)\]
Now, coming to the question part, we need to find the second and third derivative of the function. So, differentiating equation (3) again
\[\dfrac{d}{dx}\left( \dfrac{dy}{dx} \right)=\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=\dfrac{d}{dx}(\cos x)=-\sin x\]
(Differentiation of \[\cos x\] can also be proved by first principle method of differentiation).
Hence, \[\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=-\sin x-(4)\]
Now, for the third derivative of the given function; differentiate equation (4) again, as: -
\[\begin{align}
  & \dfrac{d}{dx}\left( \dfrac{{{d}^{2}}y}{d{{x}^{2}}} \right)=\dfrac{{{d}^{3}}y}{d{{x}^{3}}}=\dfrac{d}{dx}\left( -\sin x \right) \\
 & \dfrac{{{d}^{3}}y}{d{{x}^{3}}}=\dfrac{d}{dx}\left( -\sin x \right) \\
\end{align}\]
Using equation (3) as \[\dfrac{d}{dx}\left( \sin x \right)=\cos x\]we can get,
\[\dfrac{{{d}^{3}}y}{d{{x}^{3}}}=-\cos x-(5)\]
Therefore, derivative of \[\sin x\] is \[\cos x\]and second and third derivative of \[\sin x\] are
 (\[-\sin x\]) and (\[-\cos x\]) respectively from equation (4) and (5).
Note: One can go wrong while putting the values of \[\dfrac{d}{dx}(\cos x)\] and \[\dfrac{d}{dx}(\sin x)\]. Student can write \[\dfrac{d}{dx}(\sin x)=-\cos x\]or \[\dfrac{d}{dx}(\cos x)=\sin x\](confusion).
We can prove \[\dfrac{d}{dx}(\sin x)=\cos x\]by first principle method as written in the
 solution. Let us prove it.
We have formula for first principle method as: -
\[{{f}^{'}}=\underset{x\to c}{\mathop{\lim }}\,\dfrac{f(x)-f(c)}{x-c}\]
\[f(x)=\sin x\]and \[f(c)=\sin c\]
\[\underset{x\to c}{\mathop{\lim }}\,\dfrac{\sin x-\sin c}{x-c}-(1)\]
We have formula for \[\sin C-\sin D\]as
\[\sin C-\sin D=2\sin \dfrac{C-D}{2}\cos \dfrac{C+D}{2}\]
Hence,
\[\sin x-\sin c=2\sin \dfrac{x-c}{2}\cos \dfrac{x+c}{2}\]
Putting above value in equation (1)
\[\begin{align}
  & {{f}^{'}}(c)=\underset{x\to c}{\mathop{\lim }}\,\dfrac{2\sin \left( \dfrac{x-c}{2} \right)\cos

 \left( \dfrac{x+c}{2} \right)}{x-c} \\
 & f'(c)=\underset{x\to c}{\mathop{\lim }}\,\dfrac{2\sin \dfrac{\left( x-c \right)}{2}}{2\left( \dfrac{x-c}{2} \right)}\cos \left( \dfrac{x+c}{2} \right) \\
 & {{f}^{'}}(c)=\underset{x\to c}{\mathop{\lim }}\,\dfrac{\sin \dfrac{\left( x-c \right)}{2}}{\left(
 \dfrac{x-c}{2} \right)}\underset{x\to c}{\mathop{\lim }}\,\cos \left( \dfrac{x+c}{2} \right) \\
 & {{f}^{'}}(c)=1\times \cos C=\cos C\left( \because \underset{x\to 0}{\mathop{\lim }}\,\dfrac{\sin x}{x}=1 \right) \\
\end{align}\]
Hence proved