If we have a trigonometric function as $f(x)=\sin \left[ {{\pi }^{2}} \right]x+\cos \left[ -{{\pi }^{2}} \right]x$ , where [ . ] is greatest integer function, then $f'(x)$ is
( a ) 9sinx + 9cosx
( b ) 9cos9x – 10sin10x
( c ) 0
( d ) -1

Hint: First we will find the values of $\left[ {{\pi }^{2}} \right]$ and $\left[ -{{\pi }^{2}} \right]$, then substitute the values of $\left[ {{\pi }^{2}} \right]$ and $\left[ -{{\pi }^{2}} \right]$ in function $f(x)=\sin \left[ {{\pi }^{2}} \right]x+\cos \left[ -{{\pi }^{2}} \right]x$. Then using chain rule of differentiation we will find out the value of f ’ ( x ).

Complete step-by-step solution:
Before we start the question, let us see what the greatest integer function is and how do we find the value of the greatest integer function.
Let us see what is the greatest integer function and what are its properties.
Function y = [ x ] is called greatest integer function which means the greatest integer less than or equals to x. also, if n belongs to set of integer, then y = [ x ] = n if $n\le x< n+1$ that is $x\in [n,n+1)$ .
For example if we put x = -3.1 in y = [ x ], then y = - 4 and if we put x = 0.2 in y = [ x ], then y = 0.
Now, we know that ${{\pi }^{2}}={{(3.14)}^{2}}=9.85$ and $-{{\pi }^{2}}=-{{(3.14)}^{2}}=-9.85$.
So, value of $\left[ {{\pi }^{2}} \right]=\left[ 9.85 \right]=9$ and $\left[ -{{\pi }^{2}} \right]=\left[ -9.85 \right]=-10$.
So, we can re – write function $f(x)=\sin \left[ {{\pi }^{2}} \right]x+\cos \left[ -{{\pi }^{2}} \right]x$as $f(x)=\sin 9x+\cos (-10)x$
Also, cos ( - x ) =cos x.
So, $f(x)=\sin 9x+\cos 10x$
So, $f'(x)=\dfrac{d}{dx}\left( \sin 9x+\cos 10x \right)$
We know that, \[\dfrac{d}{dx}\left( A(x)+B(x) \right)=\dfrac{d}{dx}A(x)+\dfrac{d}{dx}B(x)\]
So, \[\dfrac{d}{dx}\left( \sin 9x+\cos 10x \right)=\dfrac{d}{dx}(\sin 9x)+\dfrac{d}{dx}(\cos 10x)\]
We know that $\dfrac{d}{dx}(\sin ax)=a\cdot \cos ax$ and \[\dfrac{d}{dx}(cosax)=-a\sin ax\].
So, $\dfrac{d}{dx}(\sin 9x)=9\cdot \cos 9x$ and \[\dfrac{d}{dx}(cos10x)=-10\sin 10x\].
$f'(x)=\dfrac{d}{dx}\left( \sin 9x+\cos 10x \right)=9\cos 9x10\sin 10x$
Hence, option ( b ) is true.

Note: One must know the definition of greatest integer function and how to find the value of greatest integer function for any real value of x. Always remember that $\dfrac{d}{dx}(\sin ax)=a\cdot \cos ax$ and \[\dfrac{d}{dx}(cosax)=-a\sin ax\] and \[\dfrac{d}{dx}\left( A(x)+B(x) \right)=\dfrac{d}{dx}A(x)+\dfrac{d}{dx}B(x)\].