3.3 Differentiating Inverse Functions

• The domain of f(x) is the range of f⁻¹(x).

  True
• What is the composition property of inverse functions?

  $f(f^( - 1)(x)) =$$x$
• The domain of f(x) becomes the range of f⁻¹(x)
• The derivative of f⁻¹(x) is given by the formula: $\frac{d}{dx}f^( - 1)(x) =$$\frac{1}{f'(f^( - 1)(x))}$, which relates it to the derivative of f(x)
• The inverse function derivative formula relates the derivative of f⁻¹(x) at a point x to the reciprocal of f'(x) at f⁻¹(x).


  True
• What is the derivative of f⁻¹(x) at x = 2 for f(x) = x³ + 1?

  $\frac{1}{3}$
• The derivative of the original function in the inverse derivative formula is denoted as $f'(x)$, and it represents the rate of change of $f(x)$.
• Match the components with their descriptions:

  $f'(x)$ ↔️ Derivative of the original function

  $f^( - 1)(x)$ ↔️ Inverse function of f(x)

  $\frac{d}{dx}f^( - 1)(x)$ ↔️ Derivative of the inverse function
• Inverse functions map y back to x
• The graph of f⁻¹(x) is a reflection of f(x) over the line y = x
• Inverse functions exist for all functions.

  False
• What is the composition of inverse functions?

  $f(f^( - 1)(x)) =$$x$
• What is the derivative formula for inverse functions?

  $\frac{d}{dx}f^( - 1)(x) =$$\frac{1}{f'(f^( - 1)(x))}$
• Steps to find the derivative of f⁻¹(x) at x = 2 for f(x) = x³ + 1

  1️⃣ Find the derivative of f(x): f'(x) = 3x²

  2️⃣ Evaluate f⁻¹(2): f⁻¹(2) = 1

  3️⃣ Use the inverse function derivative formula: $\frac{d}{dx}f^( - 1)(2) =$$\frac{1}{f'(1)}$

  4️⃣ Substitute f'(1): $\frac{d}{dx}f^( - 1)(2) =$$\frac{1}{3(1)^{2}}$

  5️⃣ Simplify: $\frac{d}{dx}f^( - 1)(2) =$$\frac{1}{3}$
• The domain of f⁻¹(x) is the range of f(x)
• What is the derivative of f⁻¹(x) if f(x) = x³?

  $\frac{1}{3x^(2 / 3)}$
• Inverse functions "undo" the operation of another function.

  True
• What is the derivative of the inverse function f^( - 1)(x)</latex>?

  $\frac{1}{f'(f^( - 1)(x))}$
• If $f(x) =$$x^{3} +$$2$ and f^( - 1)(x) = \sqrt[3]{x - 2}</latex>, find $\frac{d}{dx}f^( - 1)(x)$ at x = 3.


  $\frac{1}{3}$
• What is the formula to differentiate simple inverse functions?

  \frac{1}{f'(f^( - 1)(x))}</latex>
• The derivative formula for inverse functions is $\frac{d}{dx}f^( - 1)(x) =$$\frac{1}{f'(f^( - 1)(x))}$
• The first step in differentiating a composite inverse function is to find the original function f(x) and its derivative f'(x)


  True
• The second step in differentiating a composite inverse function is to find the inverse function f^(-1)(x)


  True
• The derivative formula for inverse functions is \frac{d}{dx}f^( - 1)(x) = \frac{1}{f'(f^( - 1)(x))}</latex>


  True
• In the derivative formula, f'(x) is the derivative of the original function f(x).

  True
• The notation $f^( - 1)(x)$ represents the inverse
• What is the inverse function of $f(x) =$$x^{3} +$$1$?


  $\sqrt[3]{x - 1}$
• The derivative of the inverse function \frac{d}{dx}f^( - 1)(x)</latex> for $f(x) =$$x^{3} +$$1$ is $\frac{1}{3(x - 1)^(2 / 3)}$


  True
• The graph of f⁻¹(x) is a reflection of f(x) over the line y = x.


  True
• What is the composition property of inverse functions?

  $f(f^( - 1)(x)) =$$x$
• Match the property with its corresponding description:

  Definition ↔️ Maps x to y

  Domain and Range ↔️ The domain of $f(x)$ becomes the range of $f^( - 1)(x)$

  Composition ↔️ $f(f^( - 1)(x)) =$$x$

  Differentiation ↔️ $\frac{1}{f'(f^( - 1)(x))}$
• Steps to use the derivative formula for inverse functions:

  1️⃣ Find the derivative of $f(x)$, denoted as $f'(x)$

  2️⃣ Evaluate $f^( - 1)(x)$

  3️⃣ Plug $f^( - 1)(x)$ into $f'(x)$ to get $f'(f^( - 1)(x))$

  4️⃣ Calculate the reciprocal of $f'(f^( - 1)(x))$
• $f^( - 1)(x)$ is the inverse function of $f(x)$.


  True
• Match the function with its inverse and derivative at x = 2 or x = 1:

  $f(x) =$$x^{3} +$$1$ ↔️ $f^( - 1)(x) =$$\sqrt[3]{x - 1}$

  $f(x) =$$2x - 5$ ↔️ $f^( - 1)(x) =$$\frac{x + 5}{2}$
• Steps to differentiate a composite inverse function

  1️⃣ Find the original function f(x) and its derivative f'(x)

  2️⃣ Find the inverse function f^(-1)(x)

  3️⃣ Evaluate f^(-1)(x) at the given point

  4️⃣ Plug f^(-1)(x) into f'(x) to get f'(f^(-1)(x))

  5️⃣ Take the reciprocal of f'(f^(-1)(x))
• In the derivative formula, the term f'(x) represents the derivative of the original function.
• To find the derivative of the inverse function, you take the reciprocal of f'(f^(-1)(x)).
• Inverse functions "undo" the operation of the original function.
• What does $f'(x)$ represent in the derivative formula for inverse functions?


  Derivative of f(x)
• Steps to use the derivative formula for inverse functions

  1️⃣ Find f'(x)</latex>

  2️⃣ Evaluate $f^( - 1)(x)$

  3️⃣ Plug $f^( - 1)(x)$ into $f'(x)$

  4️⃣ Calculate the reciprocal