# How do you implicitly differentiate #-3=cos(y-x)/x#?

##### 1 Answer

Mar 16, 2016

#### Explanation:

Differentiate both sides w.r.t.

#frac{"d"}{"d"x}(3) = frac{"d"}{"d"x}( cos(y-x)/x )#

#0 = frac{ xfrac{"d"}{"d"x}( cos(y-x) ) - cos(y-x)frac{"d"}{"d"x}(x) }{x^2}#

#0 = x(-sin(y-x))frac{"d"}{"d"x}(y-x) - cos(y-x)#

#0 = xsin(y-x)(frac{"d"y}{"d"x}-1) + cos(y-x)#

Now, we just have to make

#xsin(y-x) - cos(y-x) = xsin(y-x)frac{"d"y}{"d"x}#

#frac{"d"y}{"d"x} = frac{xsin(y-x) - cos(y-x)}{xsin(y-x)}#

#= 1 - cot(y-x)/x#