3.2 Implicit Differentiation

• What is implicit differentiation used for?

  Finding derivatives where y is not explicitly defined in terms of x
• The first step in implicit differentiation is to differentiate both sides of the equation with respect to x
• What is the general form of a linear equation in two variables?

  $ax +$$by =$$c$
• In explicit differentiation, is the chain rule needed for terms involving y?

  No
• When differentiating terms involving y, the chain rule requires multiplying by dy/dx.


  True
• What are the steps to differentiate both sides of an equation in implicit differentiation?

  Identify variables, apply rules, treat y, simplify
• When differentiating x^2 + y^2 = 25 with respect to x, the term d/dx[y^2] becomes 2y(dy/dx).


  True
• The chain rule is essential in implicit differentiation when dealing with the dependent variable y.
• The chain rule is essential in implicit differentiation for terms involving y.


  True
• Steps for implicit differentiation

  1️⃣ Differentiate both sides of the equation with respect to x

  2️⃣ Apply the chain rule to terms involving y

  3️⃣ Collect all terms with dy/dx on one side

  4️⃣ Solve for dy/dx
• When starting implicit differentiation, both sides of the equation must be differentiated with respect to x.


  True
• The exponential form of an equation in two variables is given by $y =$$ab^{x}$
• What is the result of differentiating x^2 + y^2 = 25 with respect to x?

  $2x +$$2y(dy / dx) =$$0$
• When differentiating terms involving y in implicit differentiation, you must apply the chain rule.
• What are the steps to differentiate both sides of an equation in implicit differentiation?

  Identify variables, apply rules, treat y, simplify
• The chain rule states that for a function f(g(x)), the derivative is f'(g(x)) * g'(x).

  True
• When differentiating x^2 + y^2 = 25 with respect to x, the chain rule is applied to the term 2y(dy/dx).


  True
• In implicit differentiation, y is implicitly defined by an equation.
• The chain rule is not needed in explicit differentiation.


  True
• An example of a linear equation in two variables is 2x + 3y = 6.
• What do equations in two variables define relationships between?

  Two variables
• A linear equation is defined as ax + by = c
• The ordered pair (3, 0) satisfies the equation 2x + 3y = 6.


  True
• When differentiating equations in implicit differentiation, use the appropriate differentiation rules such as the power rule, chain rule, product rule, or quotient rule.
• The equation x^2 + y^2 = 25 represents a circle.

  True
• What is the first step in implicit differentiation?

  Identify the variables
• What is the third step in implicit differentiation?

  Treat y as a function of x
• The final step in implicit differentiation is to solve for \frac{dy}{dx}
• The derivative of $x^{2}$ with respect to x is 2x
• The derivative $\frac{dy}{dx}$ for the equation $x^{2} +$$y^{2} =$$25$ is $- \frac{x}{y}$.


  True
• The chain rule states that $\frac{d}{dx}[f(y)] =$$f'(y) \cdot \frac{dy}{dx}$, where $f'(y)$ is the derivative of f(y) with respect to y
• When differentiating $y^{3}$ with respect to x, the result is $3y^{2} \cdot \frac{dy}{dx}$, where $\frac{dy}{dx}$ is the derivative of y with respect to x
• When solving for $\frac{dy}{dx}$, all terms with dy/dx must be collected on one side of the equation.


  True
• The equation x^2 + y^2 = 25</latex> is an example of a circle
• Recognizing y as a function of x is essential in applying the chain rule during implicit differentiation.


  True
• What is the form of the chain rule in implicit differentiation when y is a function of x?

  \frac{d}{dx}f(y) = f'(y)\cdot \frac{dy}{dx}</latex>
• What is implicit differentiation used for?

  Finding derivatives of implicitly defined functions
• Steps for implicit differentiation

  1️⃣ Differentiate both sides with respect to x

  2️⃣ Apply the chain rule to terms involving y

  3️⃣ Collect all terms with dy/dx on one side

  4️⃣ Solve for dy/dx
• What is the independent variable in an equation in two variables?

  The variable that can take any value
• The ordered pair (3, 0) satisfies the equation 2x + 3y = 6.


  True