Let $f(x)=x^4\mathrm{\&}g(x)=6x^4+5$. Find $(f\circ g{)}^{\prime }(1)$.
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Step 1 ：Given functions are $f(x)=x^4$ and $g(x)=6x^4+5$.

Step 2 ：We need to find the derivative of the composition of these functions, $(f\circ g)(x)$, at $x=1$.

Step 3 ：First, find the derivatives of $f(x)$ and $g(x)$. The derivative of $f(x)$ is $f^{\prime }(x)=4x^3$ and the derivative of $g(x)$ is $g^{\prime }(x)=24x^3$.

Step 4 ：Next, substitute $g(x)$ into $f^{\prime }(x)$ to get $f^{\prime }(g(x))=4(g(x){)}^3=4(6x^4+5{)}^3$.

Step 5 ：Then, multiply $f^{\prime }(g(x))$ by $g^{\prime }(x)$ to get $(f\circ g{)}^{\prime }(x)=f^{\prime }(g(x))\cdot g^{\prime }(x)=4(6x^4+5{)}^3\cdot 24x^3$.

Step 6 ：Finally, evaluate this expression at $x=1$ to get $(f\circ g{)}^{\prime }(1)=4(6(1{)}^4+5{)}^3\cdot 24(1{)}^3=127776$.

Step 7 ：So, the derivative of the composition of the functions $f$ and $g$ at $x=1$ is $\boxed{{\displaystyle 127776}}$.