3.2 Implicit Differentiation

• What is the key difference between implicit and explicit functions?

  Variable isolation
• Implicit functions are defined by equations of the form f(x, y) = 0
• Implicit differentiation is required to find $\frac{dy}{dx}$ in implicit functions.
• Give an example of an explicit function.

  $y =$$x^{2} +$$3$
• <question_start>x^{2} + y^{2} = 9</latex> is an example of an implicit
• Understanding the distinction between explicit and implicit functions is essential for implicit differentiation.
• What is an explicit function in terms of variable isolation?

  Dependent variable is isolated
• What differentiation method is used for explicit functions?

  Direct differentiation
• What is an implicit function in terms of variable isolation?

  Dependent variable is not isolated
• What differentiation method is required for implicit functions?

  Implicit differentiation
• $y =$$x^{2} +$$3$ is an explicit function.
• $x^{2} +$$y^{2} =$$9$ is an implicit function.
• Why is implicit differentiation necessary for certain equations?

  Dependent variable cannot be isolated
• Steps to differentiate an implicit function

  1️⃣ Differentiate both sides with respect to $x$, apply chain rule to $y$

  2️⃣ Rearrange the equation to isolate $\frac{dy}{dx}$

  3️⃣ Express $\frac{dy}{dx}$ in terms of $x$ and $y$
• An implicit function requires implicit differentiation to find $\frac{dy}{dx}$.
• What does implicit differentiation involve when $y$ cannot be isolated?


  Differentiating and rearranging
• What rule must be applied to terms involving $y$ during implicit differentiation?


  Chain rule
• Isolating \frac{dy}{dx}</latex> is a step in implicit differentiation.
• Find $\frac{dy}{dx}$ for $x^{3} +$$y^{3} =$$8$ using implicit differentiation.


  $\frac{dy}{dx} =$$- \frac{x^{2}}{y^{2}}$
• The derivative $\frac{dy}{dx}$ at a point gives the slope of the tangent line to the curve at that point.