Find y' if y = ln⁡|sec⁡ x + tan⁡ x |*sec⁡xsecx tanxsec^2x tan xsec^2 x

To find y', we need to differentiate the given expression with respect to x.

Let's start by using the chain rule.

First, we differentiate the outer function ln|secx + tanx| with respect to its argument, which is secx + tanx.

The derivative of ln|u| with respect to u is 1/u.

So, the derivative of ln|secx + tanx| with respect to secx + tanx is 1/(secx + tanx).

Next, we need to differentiate the inner function secx + tanx.

The derivative of secx is secx tanx, and the derivative of tanx is sec^2x.

Using the sum rule, the derivative of secx + tanx is secx tanx + sec^2x.

Now, we can multiply the derivative of the outer function (1/(secx + tanx)) by the derivative of the inner function (secx tanx + sec^2x) to find the derivative of y.

y' = (1/(secx + tanx)) * (secx tanx + sec^2x)

Simplifying further, we can rewrite y' as:

y' = secx tanx + sec^2x / (secx + tanx)

Therefore, the derivative of y with respect to x is secx tanx + sec^2x / (secx + tanx).