3.5 Selecting Procedures for Calculating Derivatives

• For which type of function is implicit differentiation used?

  Implicit functions
• Why is it crucial to identify the type of function when calculating derivatives?

  To apply the correct rule
• What is the derivative of \(f(x) = e^x\)?

  $e^{x}$
• Match the function type with its derivative rule:

  Polynomial ↔️ Power Rule

  Trigonometric ↔️ Derivatives of Sine and Cosine

  Exponential ↔️ Exponential Rule

  Logarithmic ↔️ Logarithmic Rule

  Composite ↔️ Chain Rule
• What is the derivative of \(f(x) = e^x\)?

  $e^{x}$
• What is the derivative of \(f(x) = \sin(e^x)\) using the chain rule?

  $\cos(e^{x}) \cdot e^{x}$
• The derivative of \(f(x) = 3x^4\) using the Power Rule is 12x^3
• What is the derivative of \(\cos(x)\)?

  -\sin(x)
• The Chain Rule is used for composite functions.

  True
• The Power Rule states that the derivative of \(ax^n\) is \(nax^{n-1}\).


  True
• The derivative of \(\ln(x)\) is \(1/x\).


  True
• The derivative of \(ax^n\) using the Power Rule is \(nax^{n-1}\).

  True
• The derivative of \(\ln(x)\) is \(1/x\).

  True
• The Chain Rule states that \(f'(x) = g'(h(x)) \cdot h'(x)\).


  True
• The Chain Rule requires multiplying the derivatives of the inner and outer functions.


  True
• Implicit differentiation is used when the relationship between x and y is explicitly defined.

  False
• Applying the Chain Rule to \(f(x) = \sin(e^x)\) gives \(f'(x) = \cos(e^x) \cdot e^x\)


  True
• What is the derivative of \(x^2 + y^2 = 25\) with respect to x using implicit differentiation?

  -\frac{x}{y}</latex>
• Match the function type with its differentiation rule:

  Polynomial ↔️ Power Rule

  Trigonometric ↔️ Derivatives of Sine and Cosine

  Exponential ↔️ Exponential Rule

  Logarithmic ↔️ Logarithmic Rule

  Composite ↔️ Chain Rule
• Once you identify the type of function, you can choose the appropriate differentiation rule


  True
• The Chain Rule states that if \(f(x) = g(h(x))\), then \(f'(x) = g'(h(x)) \cdot h'(x)\), which applies to composite functions.
• The Chain Rule states that for a composite function f(x) = g(h(x)), the derivative f'(x) is equal to g'(h(x)) multiplied by h'(x)
• The chain rule is applied when differentiating a composite function.
• The final step in calculating derivatives is to simplify the expression.
• The derivative of \(f(x) = \sin(x)\) is cos(x).
• The chain rule states that if \(f(x) = g(h(x))\), then \(f'(x) = g'(h(x)) \cdot h'(x)\), where h'(x) is the derivative of the inner function.
• The derivative of \(f(x) = \cos(x)\) is -sin(x).
• The derivative of \(f(x) = \cos(x)\) is -sin(x).
• What is the differentiation rule for a polynomial function in the form \(f(x) = ax^n\)?

  Power Rule
• The derivative of \(\sin(x)\) is \(\cos(x)\).


  True
• What is the derivative of \(f(x) = \ln(x)\)?

  \(1/x\)
• What is the primary step when calculating derivatives?

  Identify the type of function
• What is the derivative of \(e^x\)?

  \(e^x\)
• What is the first step in calculating a derivative after identifying the function type?

  Choose the appropriate rule
• What is the derivative of \(e^x\)?

  \(e^x\)
• The derivative of \(f(x) = \sin(e^x)\) using the Chain Rule is \(\cos(e^x) \cdot e^x\)
• In the Chain Rule, \(g'(h(x))\) represents the derivative of the outer function evaluated at the inner function.
• The derivative of the outer function in \(f(x) = \sin(e^x)\) is \(\cos(x)\)
• The derivative of \(g(x) = \sin(x)\) is \(g'(x) = \cos(x)\), which is a basic trigonometric rule
• Steps for implicit differentiation

  1️⃣ Differentiate both sides of the equation with respect to x, treating y as a function of x

  2️⃣ Solve for the derivative dy/dx