MathematicsIf f(x) = x1+x and g (x) = f (f(x)) then g'(x) =

If f(x) = $\frac{x}{1 + x}$ and g (x) = f (f(x)) then g'(x) =

A
$\frac{1}{{(2 x + 3)}^{2}}$
B
$\frac{1}{{(x + 1)}^{2}}$
C
$\frac{1}{x^{2}}$
D
$\frac{1}{{(2 x + 1)}^{2}}$

Solution:

Given $f (x) = \frac{x}{1 + x}$ and $g (x) = f (f (x))$

Hence, $g (x) = \frac{\frac{x}{1 + x}}{1 + \frac{x}{1 + x}} = \frac{x}{1 + 2 x}$

Use quotient rule to differentate

$\begin{array}{rcl} g' (x) & = & \frac{1 + 2 x - x (2)}{{(1 + 2 x)}^{2}} \\ = & \frac{1}{{(2 x + 1)}^{2}} \end{array}$

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