Question

Consider the following.
y = tan(cos(x))
(a) Write the composite function in the form f(g(x)) by identifying the inner function u = g(x) and the outer function y = f(u).
u = g(x) = ?
y = f(u) = ?

(b) Find the derivative dy/dx.
dy/dx = ?

Answer 1

Well, the inner function is u=cos(x) and the outer function y=tan(u) Because if you plug it in, you get the same thing you started with. The derivative is using the gain rule or:
u'f'(u). So u'=-sin(x) and y'=sec^2(u). That gives: -sin(x)sec^2(cos(x))

Answer 2

u = g(x) = cos x
y = f(u) = tanu

d/dx cos x = -sinx
d/du tan u = sec^2 u

then d/du tan(cosx) = sec^2(cosx)(-sin x)

Answer 3

"...is using the CHAIN rule..."

Answer 4

g(x)=cosx
f(u)=tanu

dy/dx=sec2(cosx)*-sinx