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Solution for Let G (X) Be the Inverse of an Invertible Function F (X) Which is Derivable at X = 3. If F (3) = 9 and F' (3) = 9, Write the Value of G' (9). - CBSE (Science) Class 12 - Mathematics

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Question

Let g (x) be the inverse of an invertible function f (x) which is derivable at x = 3. If f (3) = 9 and f' (3) = 9, write the value of g' (9).

Solution

\[\text { We have }, f\left( 3 \right) = 9 , f'\left( 3 \right) = 9\]
\[\text { and g }\left( x \right) = f^{- 1} \left( x \right)\]
\[ \Rightarrow \left( gof \right)\left( x \right) = x\]
\[ \Rightarrow g\left\{ f\left( x \right) \right\} = x\]

\[\Rightarrow \frac{d}{dx}\left[ g\left\{ f\left( x \right) \right\} \right] = 1\]
\[ \Rightarrow g'\left\{ f\left( x \right) \right\}\frac{d}{dx}\left\{ f\left( x \right) \right\} = 1\]
\[ \Rightarrow g'\left\{ f\left( x \right) \right\} \times f'\left( x \right) = 1\]
\[\text { Puting } x = 3, \text { we get }, \]
\[g'\left\{ f\left( 3 \right) \right\} \times f'\left( 3 \right) = 1\]
\[ \Rightarrow g'\left( 9 \right) \times 9 = 1 \left[ \because f\left( 3 \right) = 9 , f'\left( 3 \right) = 9 \right]\]
\[ \Rightarrow g'\left( 9 \right) = \frac{1}{9}\]

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Solution Let G (X) Be the Inverse of an Invertible Function F (X) Which is Derivable at X = 3. If F (3) = 9 and F' (3) = 9, Write the Value of G' (9). Concept: Simple Problems on Applications of Derivatives.
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