Is this right for the differential of e^secx
secx e^secx?
Yup. (As long as you mean $\displaystyle e^{\sec x} \sec x \tan x$ and not e raised to all of the rest.)
Straight continuous lines differentiate downwards with respect to x, the straight dashed line with respect to the dashed balloon expression. So the forking network on the right satisfies the chain rule - the general pattern being...
Don't integrate - balloontegrate!
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