# How do you find the derivative of #tan(x/y)=x+y#?

##### 2 Answers

#### Explanation:

#"differentiate "tan(x/y)" using the "color(blue)"chain rule"#

#rArrd/dx(tan(x/y))#

#=sec^2(x/y)xxd/dx(x/y)#

#"differentiate "x/y" using the "color(blue)"quotient rule"#

#=sec^2(x/y)xx(y-x.dy/dx)/y^2#

#=(ysec^2(x/y)-xsec^2(x/y)dy/dx)/y^2#

#"returning to the original"#

#(ysec^2(x/y)-xsec^2(x/y)dy/dx)/y^2=1+dy/dx#

#rArrysec^2(x/y)-xsec^2(x/y)dy/dx=y^2+y^2dy/dx#

#rArrdy/dx(-xsec^2(x/y)-y^2)=y^2-ysec^2(x/y)#

#rArrdy/dx=-(y^2-ysec^2(x/y))/(xsec^2(x/y)+y^2)#

#### Explanation:

You need the chain rule on the tangent part:

Distribute on the left side:

Move everything with a

Factor out the

Get

Get a common denominator in both the numerator and the denominator:

Simplify the complex fraction (I think of it as keep, change, turn):

I factored the negative out of the denominator and distributed it to the numerator:

I decided to factor the common