Suppose $f^{\prime}(x)=\sin \left(5 x^{2}\right)$
\[
\begin{array}{l}
\frac{d}{d x} f(2 x)= \\
\frac{d}{d x} f\left(7 x^{6}\right)=
\end{array}
\]

Step 1 ：Given that the derivative of the function $f(x)$ is $f'(x) = \sin(5x^2)$

Step 2 ：We are asked to find the derivative of the function at $2x$ and $7x^6$

Step 3 ：We can use the chain rule to find these derivatives. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function

Step 4 ：For $f(2x)$, the outer function is $f(x)$ and the inner function is $2x$. So, the derivative is $2\sin(20x^2)$

Step 5 ：For $f(7x^6)$, the outer function is $f(x)$ and the inner function is $7x^6$. So, the derivative is $42x^5\sin(245x^{12})$

Step 6 ：\(\boxed{\frac{d}{d x} f(2 x)= 2\sin(20x^2)}\)

Step 7 ：\(\boxed{\frac{d}{d x} f\left(7 x^{6}\right)= 42x^5\sin(245x^{12})}\)