Let $y=\tan (5x+2)$.
Find the differential $dy$ when $x=5$ and $dx=0.4$
Find the differential $dy$ when $x=5$ and $dx=0.8$

Step 1 ：Given the function $y=\tan (5x+2)$, we need to find the differential $dy$ when $x=5$ and $dx=0.4$, and when $x=5$ and $dx=0.8$.

Step 2 ：The differential of a function $y=f(x)$ is given by $dy=f^{\prime }(x)dx$. So, we first need to find the derivative of the function $y=\tan (5x+2)$.

Step 3 ：The derivative of $y=\tan (5x+2)$ is $dy/dx=5{\sec }^2(5x+2)$.

Step 4 ：Substitute $x=5$ into the derivative to get $dy/dx=5{\sec }^2(27)$.

Step 5 ：Then, substitute $dx=0.4$ into the differential equation to get $dy=5{\sec }^2(27)\ast 0.4$.

Step 6 ：Similarly, substitute $dx=0.8$ into the differential equation to get $dy=5{\sec }^2(27)\ast 0.8$.

Step 7 ：Finally, we can simplify these expressions to get the final answers.

Step 8 ：$\boxed{{\displaystyle dy=2.0+2.0{\tan }^2(27)}}$ when $x=5$ and $dx=0.4$.

Step 9 ：$\boxed{{\displaystyle dy=4.0+4.0{\tan }^2(27)}}$ when $x=5$ and $dx=0.8$.