2 Explicit Differentiation You have been taught to differentiate functions in explicit form, meaning y is defined in terms of x.Examples:The derivative isWhenever you can solve for y in terms of x, do so.
3 Explicit Differentiation Example: FindWhenever possible, rewrite in explicit form (solve for y). Then take the derivative of y with respect to x.
4 Implicit Differentiation Sometimes, however, y can’t be written in terms of x as demonstrated in the following:We need to differentiate implicitly.
5 Implicit Differentiation Remember, we are differentiating with respect to x.Using the general power rule and chain rule, we haveVariables agreeSimple power rule
6 Implicit Differentiation If variables do not agree, then use the chain rule.Variables disagreeVariables disagreeVariables disagree
7 Implicit Differentiation Using Implicit Differentiation to Find dy/dx: Four Steps to SuccessDifferentiate both sides of the equation with respect to x.Get all terms containing dy/dx alone on one side of the equation.Factor out dy/dx.Solve for dy/dx by dividing both sides of the equation by the expression remaining in parentheses.