Re: derivatives of composite function.?

Question

This question has to do with finding the derivative of a composite function.

Here is the question in which I am supposed to differentiate.

y = sec^2(x) + tan^2(x)

Here is what I have figured:

y' = (2)(secx)(secx + tanx)^2

If anyone can help me at all (including many steps) it would be very nice.

Thank you.

Answer 1

y' = [sec²(x)]' + [tan²(x)]'

To differentiate h(x) = sec²(x) we must use the chain rule:

Let f(x) = x² and g(x) = sec(x)

Therefore, the derivative is h'(x) = f'(g(x))g'(x)

From this we get that:

h'(x) = [2(sec(x)][sec(x)tan(x)] = 2sec²(x)tan(x)

Now the derivative of m(x) = tan²(x) is also done by the chain rule:

m'(x) = [j((k(x))]' = j'(k(x))k'(x)

It turns out that the derivative of m(x) is m'(x) = 2tan(x)sec²(x)

Therefore,

y' = 2sec²(x)tan(x) + 2tan(x)sec²(x)

y' = 4sec²(x)tan(x)

Hope this helps!