Derivative of Implicit Functions

Implicit differentiation means differentiating both sides of an equation with respect to x, while remembering that y depends on x. Therefore, whenever a term containing y is differentiated, the factor \[\frac{dy}{dx}\] appears by the chain rule.

Find \[\frac{dy}{dx}\], if \[y + \sin y = \cos x\].

Solution: We differentiate the relationship directly with respect to \[x\], i.e.,

which implies using the chain rule

This gives  \[\frac{dy}{dx} = -\frac{\sin x}{1 + \cos y}\]

where \[y \neq (2n + 1)\pi\]

• If an equation contains both x and y and cannot be solved directly for y, it is called an implicit function.
• Implicit functions are generally written in the form:
  f(x, y) = 0
• To differentiate an implicit function, differentiate both sides with respect to x, treating y as a function of x.