When we find functions of functions, the answer is the chain rule, 99% of the times. Let's recall it: if h and k are two differentiable functions on ]a,b[, then forall x in ]a,b[
[h(k(x))]^'=h^'(k(x))k^'(x)
In this specific case
h(u)=tan(u)
k(x)=x+sec(x)
and the derivatives are
h^'(u)=1/cos^2(u)=sec^2(u)
k^'(x)=1+sin(x)/cos^2(x)=1+sin(x)/cos(x) 1/cos(x)=1+tan(x)sec(x)
So putting all together:
f^'(x)=[tan(x+sec(x))]^'=sec^2(x+sec(x))[1+tan(x)sec(x)]
Note: To derive the tangent function and the secant function, we can recall their definitions:
[tan(x)]^'=[sin(x)/cos(x)]^'= [cos(x)cos(x)-sin(x)[-sin(x)]]/cos^2(x)=(cos^2(x)+sin^2(x))/(cos^2(x))=1/(cos^2(x))=[1/cos(x)]^2=sec^2(x)
[sec(x)]^'=[1/cos(x)]^'=-(-sin(x))/cos^2(x)=sin(x)/cos(x)*1/cos(x)=tan(x)sec(x)
