Question #056a7

1 Answer
Nov 5, 2017

sec^2(x)sec2(x)

Explanation:

Rewrite the expression: 1+tanx=1+sinx/cosx1+tanx=1+sinxcosx

=d/dx (1+sinx/cosx)ddx(1+sinxcosx)

differentiate the sum term by term

= d/dx(1)+d/dx(sinx/cosx)ddx(1)+ddx(sinxcosx)

the derivative of 1 is zero

d/dx(sinx/cosx)ddx(sinxcosx)

Use the quitient rule

(d/dx((sinx))cosx-(d/dx(cosx)sinx))/(cos^(x))ddx((sinx))cosx(ddx(cosx)sinx)cosx

simplify

(cos^2(x)+sin^2(x))/(cos^2(x)cos2(x)+sin2(x)cos2(x)

use the Pythagorean identity cos^2(x)+sin^2(x)=1cos2(x)+sin2(x)=1

=(1/cos^2(x))=(1cos2(x))

=sec^2(x)=sec2(x)