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Hint: The given problem is related to the derivative of the composite function. we have to choose \[f\left( x \right)\]and \[g\left( x \right)\] such that \[f\left( x \right)\]and \[g\left( x \right)\]are functions from\[R\] to \[R\], and \[\left( fog \right)\left( x \right)={{\left( {{x}^{3}}-{{x}^{2}}+2 \right)}^{8}}\] . Then, differentiate the functions and find the value of \[{{f}^{'}}\left( 1 \right).{{g}^{'}}\left( 1 \right)\] by substituting \[1\] in place of $x$ in the expression of the derivative.

__Complete step by step answer:__

The given composite function is \[\left( fog \right)\left( x \right)={{\left( {{x}^{3}}-{{x}^{2}}+2 \right)}^{8}}\].

First, we have to choose \[f\left( x \right)\] and \[g\left( x \right)\] . But we cannot choose any random function as \[f\left( x \right)\] and \[g\left( x \right)\] . The functions are to be chosen in such a way that they satisfy two conditions.

i) \[f\left( x \right)\] and \[g\left( x \right)\] are functions from \[R\] to \[R\] , and

ii) \[\left( fog \right)\left( x \right)={{\left( {{x}^{3}}-{{x}^{2}}+2 \right)}^{8}}\]

So, we will assume \[f\left( x \right)={{x}^{8}}\] and \[g\left( x \right)={{x}^{3}}-{{x}^{2}}+2\] .

We can see that \[f\left( x \right)\] and \[g\left( x \right)\] are function from \[R\] to \[R\] and also \[\left( fog \right)\left( x \right)=f\left( g\left( x \right) \right)={{\left( {{x}^{3}}-{{x}^{2}}+2 \right)}^{8}}\] .

Both conditions are satisfied. Hence, our assumption of functions is correct.

Now, to evaluate \[{{f}^{'}}\left( 1 \right).{{g}^{'}}\left( 1 \right)\] , first we need to calculate the values of \[f'(x)\] and \[g'(x)\] .

First, we will find the value of \[f'(x)\] . To calculate the value of \[f'(x)\] , we will differentiate \[f(x)\] with respect to \[x\] .

On differentiating \[f(x)\] with respect to \[x\], we get \[{{f}^{'}}\left( x \right)=\dfrac{d}{dx}{{x}^{8}}=8{{x}^{7}}\] .

Now, we will find the value of \[g'(x)\] . To calculate \[g'(x)\] , we will differentiate \[g(x)\] with respect to \[x\] .

On differentiating \[g(x)\] with respect to \[x\] , we get

\[{{g}^{'}}\left( x \right)=\dfrac{d}{dx}\left( {{x}^{3}}-{{x}^{2}}+2 \right)=3{{x}^{2}}-2x\]

Now, we will find the values of \[f'(1)\] and \[g'(1)\] .

To evaluate the values of \[f'(1)\] and \[g'(1)\] , we will substitute \[x=1\] in the expressions of \[f'(x)\] and \[g'(x)\] .

So, on substituting \[x=1\] in \[f'(x)=8{{x}^{7}}\] , we get \[f'(1)=8\times {{(1)}^{7}}=8\] .

And, on substituting \[x=1\] in \[g'(x)=3{{x}^{2}}-2x\] , we get \[g'(x)=3{{(1)}^{2}}-2(1)\] .

\[\begin{align}

& =3-2 \\

& =1 \\

\end{align}\]

Now, we can evaluate the value of \[{{f}^{'}}\left( 1 \right).{{g}^{'}}\left( 1 \right)\] as \[{{f}^{'}}\left( 1 \right).{{g}^{'}}\left( 1 \right)=8\times 1\]

\[=8\]

Note: While choosing\[f\left( x \right)\]and \[g\left( x \right)\], make sure that the functions have the same domain as per the question. If \[f\left( x \right)\]and \[g\left( x \right)\] are chosen at random, or without considering the condition, the value of \[{{f}^{'}}\left( 1 \right).{{g}^{'}}\left( 1 \right)\] obtained will be incorrect. Hence, before choosing the functions, students should ensure that the conditions are satisfied.

The given composite function is \[\left( fog \right)\left( x \right)={{\left( {{x}^{3}}-{{x}^{2}}+2 \right)}^{8}}\].

First, we have to choose \[f\left( x \right)\] and \[g\left( x \right)\] . But we cannot choose any random function as \[f\left( x \right)\] and \[g\left( x \right)\] . The functions are to be chosen in such a way that they satisfy two conditions.

i) \[f\left( x \right)\] and \[g\left( x \right)\] are functions from \[R\] to \[R\] , and

ii) \[\left( fog \right)\left( x \right)={{\left( {{x}^{3}}-{{x}^{2}}+2 \right)}^{8}}\]

So, we will assume \[f\left( x \right)={{x}^{8}}\] and \[g\left( x \right)={{x}^{3}}-{{x}^{2}}+2\] .

We can see that \[f\left( x \right)\] and \[g\left( x \right)\] are function from \[R\] to \[R\] and also \[\left( fog \right)\left( x \right)=f\left( g\left( x \right) \right)={{\left( {{x}^{3}}-{{x}^{2}}+2 \right)}^{8}}\] .

Both conditions are satisfied. Hence, our assumption of functions is correct.

Now, to evaluate \[{{f}^{'}}\left( 1 \right).{{g}^{'}}\left( 1 \right)\] , first we need to calculate the values of \[f'(x)\] and \[g'(x)\] .

First, we will find the value of \[f'(x)\] . To calculate the value of \[f'(x)\] , we will differentiate \[f(x)\] with respect to \[x\] .

On differentiating \[f(x)\] with respect to \[x\], we get \[{{f}^{'}}\left( x \right)=\dfrac{d}{dx}{{x}^{8}}=8{{x}^{7}}\] .

Now, we will find the value of \[g'(x)\] . To calculate \[g'(x)\] , we will differentiate \[g(x)\] with respect to \[x\] .

On differentiating \[g(x)\] with respect to \[x\] , we get

\[{{g}^{'}}\left( x \right)=\dfrac{d}{dx}\left( {{x}^{3}}-{{x}^{2}}+2 \right)=3{{x}^{2}}-2x\]

Now, we will find the values of \[f'(1)\] and \[g'(1)\] .

To evaluate the values of \[f'(1)\] and \[g'(1)\] , we will substitute \[x=1\] in the expressions of \[f'(x)\] and \[g'(x)\] .

So, on substituting \[x=1\] in \[f'(x)=8{{x}^{7}}\] , we get \[f'(1)=8\times {{(1)}^{7}}=8\] .

And, on substituting \[x=1\] in \[g'(x)=3{{x}^{2}}-2x\] , we get \[g'(x)=3{{(1)}^{2}}-2(1)\] .

\[\begin{align}

& =3-2 \\

& =1 \\

\end{align}\]

Now, we can evaluate the value of \[{{f}^{'}}\left( 1 \right).{{g}^{'}}\left( 1 \right)\] as \[{{f}^{'}}\left( 1 \right).{{g}^{'}}\left( 1 \right)=8\times 1\]

\[=8\]

Note: While choosing\[f\left( x \right)\]and \[g\left( x \right)\], make sure that the functions have the same domain as per the question. If \[f\left( x \right)\]and \[g\left( x \right)\] are chosen at random, or without considering the condition, the value of \[{{f}^{'}}\left( 1 \right).{{g}^{'}}\left( 1 \right)\] obtained will be incorrect. Hence, before choosing the functions, students should ensure that the conditions are satisfied.