# What is the implicit derivative of #1=xy-cosy#?

##### 1 Answer

#### Explanation:

Implicit differentiation is basically an application of the chain rule.

#frac{"d"}{"d"x}(f(y)) = frac{"d"}{"d"y}(f(y)) frac{"d"y}{"d"x}#

#= f'(y) frac{"d"y}{"d"x}#

So in this problem, we take the derivative of

#frac{"d"}{"d"x}(1) = frac{"d"}{"d"x}(xy - cos(y))#

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

#= [yfrac{"d"}{"d"x}(x)+xfrac{"d"}{"d"x}(y)] - frac{"d"}{"d"y}(cos(y))frac{"d"y}{"d"x}#

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

Now, we just have to make

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

#= (x + sin(y))frac{"d"y}{"d"x}#

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