F(x)=ex. Sinx÷cosx

Question

Find the derivative of the equation and then find the second derivative

Answer 1

F(x)=e^xsin(x)/cos(x)=e^xtan(x),

dF/dx=e^xsec²(x)+e^xtan(x)=e^x(sec²(x)+tan(x)),

d²F/dx²=e^x(sec²(x)+tan(x))+e^x(2sec(x)sec(x)tan(x)+sec²(x)),

d²F/dx²=e^x(sec²(x)+tan(x))+e^x(2sec²(x)tan(x)+sec²(x)),

d²F/dx²=e^x(2sec²(x)+tan(x)+2sec²(x)tan(x))=e^x(2sec²(x)(1+tan(x))+tan(x)).

There are other ways to present this derivative, for example, using the identity sec²(x)=tan²(x)+1:

d²F/dx²=e^x(2tan³(x)+2tan²(x)+3tan(x)+2), which involves only e^x and tan(x).