Example 5

Find dy/dx where This problem requires three uses of the Chain Rule.

Let 

u = sin (4x + 1) + cos (4x - 1), y = √u.

Then by the Chain Rule,

Now let

v = sin (4x + 1), w = cos (4x — 1), u = v + w.

Then

We use the Chain Rule twice more to find dv/dx and dw/dx.

Finally, we combine everything to get