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Let f and g be differentiable functions on R such that fo g is the identity function. If for some a,b\epsilon \textbf{R},g^{'}(a)=5\: \: and\: \: g(a)=b then f^{'}(b) is equal to :   
Option: 1 \frac{2}{5}
Option: 2 5
Option: 3 1
Option: 4 \frac{1}{5}
 

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Rules of Differentiation (Chain Rule) -

Rules of Differentiation (Chain Rule)

Chain Rule or Derivation of Composite Function:

The chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Let f and g be functions. For all x in the domain of g for which g is differentiable at x and f is differentiable at g(x), the derivative of the composite function

h(x) = (f?g)(x) = f (g(x)) Is given by

h′(x) = f’(g(x))?g’(x)

Composites of Three or More Functions

For all values of x for which the function is differentiable, if k(x) = h(f(g(x)))Then,

k^{\prime}(x)=h'\left ( f(g(x)) \right )\cdot f'\left ( g(x) \right )\cdot g'(x)

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\\f(g(x))=x \\\Rightarrow f^{\prime}(g(x)) \cdot g^{\prime}(x)=1 \\Put \;\;x=a \\\Rightarrow f^{\prime}(g(a)) g^{\prime}(a)=1 \\\Rightarrow f^{\prime}(b) \times 5=1 \Rightarrow f^{\prime}(b)=\frac{1}{5}

Correct Option 4

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