Let $f(x)=x^{4} \& g(x)=6 x^{4}+5$. 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 \(f'(x) = 4x^{3}\) and the derivative of \(g(x)\) is \(g'(x) = 24x^{3}\).

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

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

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

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