The standard operations between two functions with domain and codomain $R$ are addition, subtraction, multiplication, division, and composition. We have rules which tell us how to differentiate the outcomes of these operations, namely, The Algebraic Differentiability Theorem, The Product Rule, The Quotient Rule, and The Chain Rule. Suppose we invent a new operation between functions, say $,$ defined by

$(fg)(x)=\frac{f(x{)}^2+1}{g(x{)}^2+1},$ for all xinR

$($a$)$ What would be the "rule" for differentiating the outcome of this operation? To be clear, the rule for differentiating the outcome of multiplication is $(fg{)}^{\text{'}}(x)=f^{\text{'}}(x)g(x)+f(x)g^{\text{'}}(x).$ Give a similar rule for $(fg{)}^{\text{'}}(x).$

$($b$)$ Use the definition of the derivative of a function to prove the equation given in part $($a$).$

Solve all parts and be extensive please.