Suppose that $$ f(4)=4, f^{\prime}(4)=-4, g(4)=4 $$, and $$ g^{\prime}(4)=2 $$. If $$ F(x)=(f \circ g)(x) $$ and $$ G(x)=(g \circ f)(x) $$, find $$ F^{\prime}(4)+G^{\prime}(4) $$. Answer

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To find $$ F^{\prime}(4) + G^{\prime}(4) $$, we will first need to apply the chain rule to differentiate $$ F(x) $$ and $$ G(x) $$.

1. **Finding $$ F^{\prime}(x) $$**:
   $$
   F(x) = (f \circ g)(x) = f(g(x))
   $$
   By the chain rule, the derivative is:
   $$
   F^{\prime}(x) = f^{\prime}(g(x)) \cdot g^{\prime}(x)
   $$
   Now, we need to evaluate this at $$ x = 4 $$:
   $$
   F^{\prime}(4) = f^{\prime}(g(4)) \cdot g^{\prime}(4)
   $$
   We know that $$ g(4) = 4 $$, so:
   $$
   F^{\prime}(4) = f^{\prime}(4) \cdot g^{\prime}(4)
   $$
   Substituting the known values:
   $$
   F^{\prime}(4) = (-4) \cdot 2 = -8
   $$

2. **Finding $$ G^{\prime}(x) $$**:
   $$
   G(x) = (g \circ f)(x) = g(f(x))
   $$
   Again, using the chain rule:
   $$
   G^{\prime}(x) = g^{\prime}(f(x)) \cdot f^{\prime}(x)
   $$
   Evaluating at $$ x = 4 $$:
   $$
   G^{\prime}(4) = g^{\prime}(f(4)) \cdot f^{\prime}(4)
   $$
   We know that $$ f(4) = 4 $$, so:
   $$
   G^{\prime}(4) = g^{\prime}(4) \cdot f^{\prime}(4)
   $$
   Substituting the known values:
   $$
   G^{\prime}(4) = 2 \cdot (-4) = -8
   $$

3. **Calculating $$ F^{\prime}(4) + G^{\prime}(4) $$**:
   $$
   F^{\prime}(4) + G^{\prime}(4) = -8 + (-8) = -16
   $$

Thus, the final answer is:
$$
\boxed{-16}
$$
$$ F^{\prime}(4) + G^{\prime}(4) = -16 $$

Suppose that \( f(4)=4, f^{\prime}(4)=-4, g(4)=4 \), and \( g^{\prime}(4)=2 \). If \( F(x)=(f \circ g)(x) \) and \( G(x)=(g \circ f)(x) \), find \( F^{\prime}(4)+G^{\prime}(4) \). Answer

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