3.5 Selecting Procedures for Calculating Derivatives

• Why is it important to recognize the type of function (composite, implicit, or inverse)?

  To choose the correct differentiation procedure
• The derivative of $f(g(x))$ is $\frac{d}{dx} f(g(x)) =$$f'(g(x)) \cdot g'(x)$, which is known as the chain
• Inverse functions reverse the roles of $x$ and $y$.


  True
• A composite function is one where an outer function operates on an inner function
• Implicit functions define $y$ explicitly in terms of $x$.


  False
• An inverse function reverses the roles of $x$ and y
• Steps to apply the chain rule for composite functions

  1️⃣ Identify the inner function g(x)</latex> and the outer function $f(u)$.

  2️⃣ Find the derivative of the outer function with respect to the inner function $f'(u)$.

  3️⃣ Find the derivative of the inner function $g'(x)$.

  4️⃣ Multiply the derivatives together and substitute the inner function back into the outer derivative.
• When applying the chain rule, the first step is to identify the inner and outer
• A composite function is one where an outer function operates on an inner function.


  True
• To differentiate an implicit function, we use the process of implicit differentiation
• What is the formula for $\frac{dy}{dx}$ in implicit differentiation?


  -\frac{F_{x}}{F_{y}}</latex>
• What is the reciprocal of the derivative of the original function called in the inverse function theorem?

  $\frac{1}{f'(f^{ - 1}(x))}$
• Match the function type with its characteristics and differentiation procedure:

  Composite ↔️ Chain rule applies | Use the chain rule

  Implicit ↔️ $y$ is not explicitly defined | Use implicit differentiation

  Inverse ↔️ Reverses roles of $x$ and $y$ | Use the inverse function theorem
• Steps to apply the chain rule for $\frac{d}{dx} f(g(x))$


  1️⃣ Identify the inner function, $g(x)$, and the outer function, $f(u)$

  2️⃣ Find the derivative of the outer function with respect to the inner function, $f'(u)$

  3️⃣ Find the derivative of the inner function, $g'(x)$

  4️⃣ Multiply the derivatives together and substitute the inner function back into the outer derivative
• The derivative of $f(g(x))$ is given by f'(g(x)) \cdot g'(x)
• The chain rule is only used for functions of the form $f(x) \cdot g(x)$


  False
• The final derivative of $\frac{d}{dx} \sin(x^{2})$ is 2x \cos(x^{2})
• What are implicit functions?

  Functions where y is not explicitly defined in terms of x
• What does the inverse function theorem allow you to find?

  The derivative of an inverse function
• The derivative of $f^{ - 1}(x)$ at $x =$$3$ for $f(x) =$$x^{3} +$$2x$ is $\frac{1}{5}$


  True
• Composite functions use the chain rule to find their derivatives.


  True
• What is the formula for implicit differentiation?

  \frac{dy}{dx} = - \frac{F_{x}}{F_{y}}</latex>
• What is the formula for the inverse function theorem?

  (f^{ - 1})'(x) = \frac{1}{f'(f^{ - 1}(x))}</latex>
• What is the derivative of \sin(x^{2})</latex> using the chain rule?

  $2x \cos(x^{2})$
• What procedure is used to differentiate implicit functions?

  Implicit differentiation
• The chain rule states that the derivative of $f(g(x))$ is $f'(g'(x)) \cdot g(x)$.


  False
• What is the derivative of $f(g(x))$?


  $f'(g(x)) \cdot g'(x)$
• Steps to apply the chain rule for $\frac{d}{dx} f(g(x))$


  1️⃣ Identify the inner function, $g(x)$, and the outer function, $f(u)$

  2️⃣ Find the derivative of the outer function with respect to the inner function, $f'(u)$

  3️⃣ Find the derivative of the inner function, $g'(x)$

  4️⃣ Multiply the derivatives together and substitute the inner function back into the outer derivative
• What is the derivative of $\sin(x^{2})$?


  $2x \cos(x^{2})$
• In an implicit function, the variable $y$ is explicitly defined in terms of $x$.


  False
• The inverse function theorem states that $(f^{ - 1})'(x) =$$\frac{1}{f'(f^{ - 1}(x))}$.theorem
• The inverse function theorem allows us to differentiate an inverse function without needing to find its explicit form.


  True
• What is the chain rule used for?

  Differentiating composite functions
• Steps to apply the chain rule

  1️⃣ Identify the inner and outer functions

  2️⃣ Find the derivative of the outer function with respect to the inner function

  3️⃣ Find the derivative of the inner function

  4️⃣ Multiply the derivatives together and substitute the inner function back into the outer derivative
• What is the formula for the chain rule?

  $\frac{d}{dx} f(g(x)) =$$f'(g(x)) \cdot g'(x)$
• Composite functions always involve an outer function and an inner function


  True
• Steps to apply implicit differentiation

  1️⃣ Differentiate both sides with respect to x

  2️⃣ Use the chain rule for terms involving y

  3️⃣ Collect terms with $\frac{dy}{dx}$

  4️⃣ Solve for $\frac{dy}{dx}$
• The derivative of x^{2} + y^{2} = 1</latex> with respect to x is $\frac{dy}{dx} =$$- \frac{x}{y}$


  True
• Steps to apply the inverse function theorem

  1️⃣ Find $f^{ - 1}(3)$

  2️⃣ Find $f'(x)$

  3️⃣ Evaluate $f'(f^{ - 1}(3))$

  4️⃣ Apply the inverse function theorem
• The type of function determines the differentiation procedure

  True