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Question
If \[f\left( x \right) = x + 1\] , then write the value of \[\frac{d}{dx} \left( fof \right) \left( x \right)\] ?
Solution
\[\text {We have }, f\left( x \right) = x + 1 \]
\[\text { Now, } \left( fof \right)\left( x \right) = f\left( f\left( x \right) \right) \]
\[ \Rightarrow \left( fof \right)\left( x \right) = f\left( x + 1 \right)\]
\[ \Rightarrow \left( fof \right)\left( x \right) = \left( x + 1 \right) + 1\]
\[ \Rightarrow \left( fof \right) = x + 2\]
\[\Rightarrow \frac{d}{dx}\left\{ \left( fof \right)\left( x \right) \right\} = \frac{d}{dx}\left( x \right) + \frac{d}{dx}\left( 2 \right)\]
\[ \Rightarrow \frac{d}{dx}\left\{ \left( fof \right)\left( x \right) \right\} = 1 + 0\]
\[ \Rightarrow \frac{d}{dx}\left\{ \left( fof \right)\left( x \right) \right\} = 1\]
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