Write the composite function in the form $f(g(x))$. [Identify the inner function $u=g(x)$ and the outer function $y=f(u)$.] (Use non-identity functions for $f(u)$ and $g(x)$.) \[ y=\left(3-x^{6}\right)^{4} \] $(f(u), g(x))=$ Find the derivative $\frac{d y}{d x}$. \[ \frac{d y}{d x}= \] Need Help? Road It Submit Answer

Step 1 ：First, we identify the inner function $u=g(x)$ and the outer function $y=f(u)$. In this case, $g(x)=3-x^{6}$ and $f(u)=u^{4}$. So, $(f(u), g(x))=(u^{4}, 3-x^{6})$.

Step 2 ：Next, we find the derivative of $y$ with respect to $x$. We use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, times the derivative of the inner function. In mathematical notation, this is $\frac{d y}{d x}=f'(g(x)) \cdot g'(x)$.

Step 3 ：We first find $f'(u)$, the derivative of $f(u)$ with respect to $u$. Since $f(u)=u^{4}$, $f'(u)=4u^{3}$.

Step 4 ：Next, we find $g'(x)$, the derivative of $g(x)$ with respect to $x$. Since $g(x)=3-x^{6}$, $g'(x)=-6x^{5}$.

Step 5 ：Now we can find $\frac{d y}{d x}$ by substituting $f'(u)$ and $g'(x)$ into the chain rule formula. We get $\frac{d y}{d x}=f'(g(x)) \cdot g'(x)=4(3-x^{6})^{3} \cdot (-6x^{5})$.

Step 6 ：Simplifying this expression, we get $\frac{d y}{d x}=-24x^{5}(3-x^{6})^{3}$.