2.8 The Product Rule

• The Product Rule is used to differentiate the product of two functions
• If \( f(x) = x^2 \) and \( g(x) = \sin(x) \), then \( f'(x) = 2x \) and \( g'(x) = \cos(x) \)


  True
• The Product Rule formula is \frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x)</latex>


  True
• What is the Product Rule used for?

  Differentiating the product of two functions
• The Product Rule can only be applied if both functions are differentiable.


  True
• In the Product Rule formula, \( u(x) \) represents the first function
• If \( f(x) = (x^2 + 1) \cdot \cos(x) \), what are the two functions being multiplied?

  \( x^2 + 1 \) and \( \cos(x) \)
• In the Product Rule table, the derivative of \( f(x) \cdot g(x) \) is \( f'(x) \cdot g(x) + f(x) \cdot g'(x) \)
• Steps to apply the Product Rule

  1️⃣ Identify the two functions in the product

  2️⃣ Differentiate each function individually

  3️⃣ Apply the Product Rule formula
• If \( u(x) = x^3 \) and \( v(x) = \sin(x) \), then \( u'(x) = 3x^2 \) and \( v'(x) = \cos(x) \)


  True
• The formula for the Product Rule is f'(x)
• The Product Rule formula can also be written as \(\frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x)\).


  True
• When using the Product Rule, the first step is to identify the two functions being multiplied.


  True
• If \( u(x) = x^3 \) and \( v(x) = \sin(x) \), what are their derivatives \( u'(x) \) and \( v'(x) \)?

  \( u'(x) = 3x^2 \) and \( v'(x) = \cos(x) \)
• When applying the Product Rule, you must find the derivatives \( u'(x) \) and \( v'(x) \) of the functions \( u(x) \) and \( v(x) \)


  True
• What is the derivative of \( x^2 \cdot \sin(x) \) using the Product Rule?

  \(2x \cdot \sin(x) + x^2 \cdot \cos(x)\)
• What is the derivative of \( (x^3 - 2x) \cdot \sin(x) \) using the Product Rule?

  \((3x^2 - 2)\sin(x) + (x^3 - 2x)\cos(x)\)
• The derivative of \( f(x) = u(x) \cdot v(x) \cdot w(x) \) is \( f'(x) = u'(x) \cdot v(x) \cdot w(x) + u(x) \cdot v'(x) \cdot w(x) + u(x) \cdot v(x) \cdot w'(x) \).


  True
• If \( u(x) = x^2 \), then \( u'(x) = 2x
• The Product Rule states that \( \frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) \), where \( u'(x) \) is the derivative of \( u(x) \) and \( v'(x) \) is the derivative of \( v(x)
• What is the derivative of \( x^2 \cdot \sin(x) \) using the Product Rule?

  \( 2x \cdot \sin(x) + x^2 \cdot \cos(x) \)
• When using the Product Rule, the first step is to identify the two functions being multiplied.


  True
• To apply the Product Rule formula, you must first identify \( u(x) \) and \( v(x) \) from the product
• When differentiating \( x \cdot \ln(x) \) using the Product Rule, the derivative of \( \ln(x) \) is \( \frac{1}{x} \), and the derivative of \( x \) is 1
• The derivative of \(f(x) = x \cdot \ln(x)\) is \(f'(x) = \ln(x) + 1
• What is the derivative of \(f(x) = x^2 \cdot \sin(x) \cdot e^x\)?

  2x\sin(x)e^x + x^2\cos(x)e^x + x^2\sin(x)e^x
• The Product Rule formula is $\frac{d}{dx} [f(x) \cdot g(x)] =$$f'(x) \cdot g(x) +$$f(x) \cdot g'(x)$


  True
• Applying the Product Rule to \( x^2 \cdot \sin(x) \) results in \( 2x \cdot \sin(x) + x^2 \cdot \cos(x) \)
• To apply the Product Rule, you must differentiate each function individually
• If \( f(x) = x^2 \) and \( g(x) = \sin(x) \), what is the derivative of their product using the Product Rule?

  $2x \cdot \sin(x) +$$x^{2} \cdot \cos(x)$
• If \( u(x) = x^2 \) and \( v(x) = \sin(x) \), what is the derivative of their product using the Product Rule?

  $2x \cdot \sin(x) +$$x^{2} \cdot \cos(x)$
• To apply the Product Rule, you must differentiate each function
• What is the key focus when applying the Product Rule?

  Differentiate each function individually
• The Product Rule formula is \(\frac{d}{dx} [u(x) \cdot v(x)]\)
• The derivative of \( x \cdot \ln(x) \) is \( \ln(x) + 1 \)


  True
• The Product Rule can be extended to differentiate the product of more than two functions
• What is \( u'(x) \) if \( u(x) = x^2 \)?

  2x
• Match the function with its derivative:

  \( u(x) \cdot v(x) \cdot w(x) \) ↔️ \( u'(x) \cdot v(x) \cdot w(x) + u(x) \cdot v'(x) \cdot w(x) + u(x) \cdot v(x) \cdot w'(x) \)

  \( u(x) \cdot v(x) \) ↔️ \( u'(x) \cdot v(x) + u(x) \cdot v'(x) \)
• Steps to apply the Product Rule

  1️⃣ Identify the two functions \( u(x) \) and \( v(x) \)

  2️⃣ Find their derivatives \( u'(x) \) and \( v'(x) \)

  3️⃣ Apply the formula \( \frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) \)
• What is the derivative of \( x^2 \cdot \sin(x) \) using the Product Rule?

  2x \sin(x) + x^2 \cos(x)