given f(1)=1, f(1)=2, g(1)=4, g(1)=17, f(4)=7, and g(4)=4, compute the derivative $\frac{d}{dx}f(g(x))|_{x =…

Explanation:

Step1: Apply chain - rule

The chain - rule states that if $y = f(g(x))$, then $\frac{dy}{dx}=f^{\prime}(g(x))\cdot g^{\prime}(x)$.

Step2: Evaluate at $x = 1$

We want to find $\left.\frac{d}{dx}[f(g(x))]\right|{x = 1}$. Substitute $x = 1$ into the chain - rule formula: $\left.\frac{d}{dx}[f(g(x))]\right|{x = 1}=f^{\prime}(g(1))\cdot g^{\prime}(1)$.

Step3: Use given values

We know that $g(1)=4$ and $g^{\prime}(1)=17$, and $f^{\prime}(4) = 7$. Since $g(1)=4$, then $f^{\prime}(g(1))=f^{\prime}(4)$. So, $\left.\frac{d}{dx}[f(g(x))]\right|_{x = 1}=f^{\prime}(4)\cdot g^{\prime}(1)$.

Step4: Calculate the result

Substitute $f^{\prime}(4)=7$ and $g^{\prime}(1)=17$ into the expression: $7\times17 = 119$.

Answer:

$119$