3.2 Implicit Differentiation

• An implicit function is defined by an explicit formula.

  False
• Match the type of function with its definition:

  Explicit function ↔️ Defined by formula $y =$$f(x)$

  Implicit function ↔️ Defined by equation $F(x,y) =$$0$
• The chain rule is applied differently in implicit and explicit differentiation.

  True
• Grouping terms containing $\frac{dy}{dx}$ is necessary to isolate it.


  True
• Steps to isolate $\frac{dy}{dx}$ in a differentiated equation


  1️⃣ Differentiate both sides of the equation with respect to $x$.

  2️⃣ Group all terms containing $\frac{dy}{dx}$ on one side.

  3️⃣ Factor out $\frac{dy}{dx}$.

  4️⃣ Divide by the factor to solve for $\frac{dy}{dx}$.
• Steps to isolate $\frac{dy}{dx}$ in a differentiated equation


  1️⃣ Differentiate both sides of the equation with respect to $x$.

  2️⃣ Group all terms containing $\frac{dy}{dx}$ on one side.

  3️⃣ Factor out $\frac{dy}{dx}$.

  4️⃣ Divide by the factor to solve for $\frac{dy}{dx}$.
• In implicit application problems, x</latex> and $y$ are related by an equation.


  True
• An explicit function can be written in the form $y =$$f(x)$.


  True
• Grouping terms containing $\frac{dy}{dx}$ is necessary to isolate it.


  True
• To solve for $\frac{dy}{dx}$, the final step is to divide
• What algebraic manipulation is used to differentiate an implicit function?

  Algebraic manipulation
• The first step in solving related rates problems is to identify variables and their rates.

  True
• What is the rate at which the water level is rising in the conical tank when it is 5 meters deep?

  $\frac{1}{2\pi}$ m/min
• Implicit functions are differentiated using standard differentiation rules.

  False
• Steps to isolate $\frac{dy}{dx}$ in a differentiated equation


  1️⃣ Group terms containing $\frac{dy}{dx}$

  2️⃣ Factor out $\frac{dy}{dx}$

  3️⃣ Divide both sides to solve for $\frac{dy}{dx}$
• Give an example of an implicit function.

  $x^{2} +$$y^{2} =$$1$
• The chain rule is applied to the dependent variable when differentiating implicit functions.


  True
• The equation $x^{2} +$$y^{2} =$$1$ defines an explicit function.


  False
• What is the result of applying the chain rule to $y^{2}$ in the equation $x^{2} +$$y^{2} =$$1$?


  $2y\frac{dy}{dx}$
• Steps to differentiate an implicit function

  1️⃣ Differentiate both sides of the equation

  2️⃣ Isolate the derivative of the dependent variable
• The chain rule is applied to the independent variable in implicit differentiation.

  False
• Match the function type with its chain rule application:

  Explicit function ↔️ Chain rule applied directly to $y =$$f(x)$

  Implicit function ↔️ Chain rule applied to terms involving $y$
• When applying the chain rule to terms involving $y$, it is necessary to treat $\frac{dy}{dx}$ as a separate term
• In an explicit function, isolating $\frac{dy}{dx}$ involves direct derivation
• Implicit differentiation is crucial for finding rates of change in problems where the relationship between variables is not explicitly defined
• Steps to solve related rates problems using implicit differentiation

  1️⃣ Differentiate both sides with respect to time $t$.

  2️⃣ Plug in known values.

  3️⃣ Solve for the desired rate of change.
• Steps to differentiate an implicit function

  1️⃣ Differentiate both sides of the equation with respect to the independent variable.

  2️⃣ Isolate the derivative of the dependent variable ($\frac{dy}{dx}$) on one side.
• In explicit functions, $\frac{dy}{dx}$ is found by direct derivation.


  True
• Steps to differentiate $x^{2} +$$y^{2} =$$1$ using implicit differentiation


  1️⃣ Differentiate: $2x +$$2y\frac{dy}{dx} =$$0$

  2️⃣ Group terms: $2y\frac{dy}{dx} =$$- 2x$

  3️⃣ Divide to solve: $\frac{dy}{dx} =$$- \frac{x}{y}$
• Match the equation type with its differentiation method:

  Explicit Function ↔️ Direct derivation

  Implicit Function ↔️ Algebraic manipulation
• What is the third step in solving related rates problems?

  Differentiate both sides
• An implicit function is defined by an equation rather than an explicit formula.


  True
• What is the first step when differentiating an implicit function using the chain rule?

  Differentiate both sides
• Steps to isolate $\frac{dy}{dx}$ in a differentiated equation


  1️⃣ Differentiate both sides of the equation with respect to $x$

  2️⃣ Group terms containing $\frac{dy}{dx}$ on one side

  3️⃣ Factor out $\frac{dy}{dx}$

  4️⃣ Divide to solve for $\frac{dy}{dx}$
• Explicit functions express the dependent variable directly in terms of the independent variable.
• What is the result of differentiating both sides of $x^{2} +$$y^{2} =$$1$?


  $2x +$$2y\frac{dy}{dx} =$$0$
• What is the first step in differentiating an implicit function?

  Differentiate both sides
• Factoring out $\frac{dy}{dx}$ is a step in isolating it.


  True
• What is an implicit function defined by?

  An equation
• What is an explicit function defined by?

  A formula