Question

# If $f\left( x \right)$ and $g\left( x \right)$ are two functions from $R$ to $R$ such that $\left( fog \right)\left( x \right)={{\left( {{x}^{3}}-{{x}^{2}}+2 \right)}^{8}}$ , then, The value of ${{f}^{'}}\left( 1 \right).{{g}^{'}}\left( 1 \right)$ is

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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.

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.