Find the derivative of f(x)=-tan(x)?

1 Answer
Feb 12, 2018

f'(x)=-sec^2(x)

Explanation:

We want to find the derivative of

f(x)=-tan(x)

Use the definition tan(x)=sin(x)/cos(x)

f(x)=-sin(x)/cos(x)

Use the quotient rule, if f(x)=(h(x))/g(x),

then f'(x)=(h'(x)g(x)-h(x)g'(x))/(h(x))^2

By the quotient rule with h(x)=sin(x) and g(x)=cos(x)

f'(x)=-((d/dx(sin(x)))cos(x)-sin(x)(d/dx(cos(x))))/cos^2(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)

=-sec^2(x)