If f(x)=g(x)+g(-x)2+2h(x)+h(-x)-1,where g and h are differentiable functions then ⨍ ‘(0)=

# If $\text{f(x)=}\frac{\mathrm{g}\left(\mathrm{x}\right)+\mathrm{g}\left(-\mathrm{x}\right)}{2}\text{+}\frac{2}{{\left[\mathrm{h}\left(\mathrm{x}\right)+\mathrm{h}\left(-\mathrm{x}\right)\right]}^{-1}}\text{,}$where g and h are differentiable functions then

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### Solution:

The given function is

This can be written as

$f\left(x\right)=\frac{g\left(x\right)+g\left(-x\right)}{2}+2\left[h\left(x\right)+h\left(-x\right)\right]$

Differentiate both sides

$f\text{'}\left(x\right)=\frac{g\text{'}\left(x\right)-g\text{'}\left(-x\right)}{2}+2\left[h\text{'}\left(x\right)-h\text{'}\left(-x\right)\right]$

Substitute $x=0$

$f\text{'}\left(0\right)=0$

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