Determine the value of d/dx at the given value of xf(z)=2z/z^2+1, z=g(x)=10x^2+x+1

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You need to evaluate f(z) in terms of x substituting g(x) for z in expression of function f(z) such that:

`f(z) = f(g(x)) = (2g(x))/(g^2(x) + 1)`

You need to substitute `10x^2+x+1`  for g(x) in equation of function such that:

`f(g(x)) = 2(10x^2+x+1)/((10x^2+x+1)^2 + 1)`

You need to differentiate with respect to x such that:

`f'(g(x)) = ((20x^2+2x+2)'*((10x^2+x+1)^2 + 1) - (20x^2+2x+2)*((10x^2+x+1)^2 + 1)')/(((10x^2+x+1)^2 + 1)^2)`

`f'(g(x)) = ((40x+2)*((10x^2+x+1)^2 + 1) - (20x^2+2x+2)*(2(10x^2+x+1)(20x+1))/(((10x^2+x+1)^2 + 1)^2)`

`f'(g(x)) = ((40x+2) - 2(10x^2+x+1)^2*(20x+1))`

Factoring out `2(20x+1) ` yields:

`f'(g(x)) = 2(20x+1)(1 - (10x^2+x+1)^2)/(((10x^2+x+1)^2 + 1)^2)`

Hence, differentiating the function f(z) under the given conditions yields `f'(g(x)) = 2(20x+1)(1 - (10x^2+x+1)^2)/(((10x^2+x+1)^2 + 1)^2).`

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