If f(x) = x^ 2 + 1 ; g(x) = x + 1 x^ 2 + 1 and h(x) = 2x - 3, the… - Vidyadip

Question

$\text{If f(x)} = \sqrt{\text{x}^{2} + 1}; \text{g(x)} = \frac{\text{x + 1}}{\text{x}^{2} + 1}$ and $\text{h(x) = 2x - 3,}$ then find $\text{f }'[\text{h }'\big\{\text{g }'(\text{x})\big\}].$

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Answer

$\text{f(x)} = \sqrt{\text{x}^{2} + 1}, \text{g(x)} = \frac{\text{x + 1}}{\text{x}^{2} + 1}, \text{h(x)} = \text{2x - 3}$
Differentiating w.r.t. “x”, we get
$\text{f'(x)} = \frac{\text{x}}{\sqrt{\text{x}^{2} + 1}}, \text{g(x)} = \frac{1 -2\text{x - x}^{2}}{(\text{x}^{2} + 1)^{2}}, \text{h'(x)} = 2$
$\therefore\text{f' (h' (g' (x)))} = \frac{2}{\sqrt{5}}$

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