Question

Question #66881

Answer 1

`d/dxtan(x) = sec^2(x)`

Explanation:

Using the quotient rule along with the derivatives for sine and cosine

• `d/dx sin(x) = cos(x)`
• `d/dx cos(x) = -sin(x)`

we have

`d/dxtan(x) = d/dxsin(x)/cos(x)`

`=(cos(x)(d/dxsin(x))-sin(x)(d/dxcos(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)`

`=(1/cos(x))^2`

`=sec^2(x)`