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Understanding the Product Rule With 3 Terms
Understanding the Product Rule With 3 Terms
Understanding the Product Rule With 3 Terms

What Is the Product Rule?

The Product Rule is a differentiation rule that applies when you need to find the derivative of the product of two or more functions. For two functions, the product rule states: ddx[u(x)v(x)]=u′(x)v(x)+u(x)v′(x)\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)This rule extends to three functions as well, allowing us to differentiate the product u(x)v(x)w(x)u(x)v(x)w(x).

The Product Rule for 3 Terms

For three functions u(x)u(x), v(x)v(x), and w(x)w(x), the derivative of their product is: ddx[u(x)v(x)w(x)]=u′(x)v(x)w(x)+u(x)v′(x)w(x)+u(x)v(x)w′(x)\frac{d}{dx}[u(x)v(x)w(x)] = u'(x)v(x)w(x) + u(x)v'(x)w(x) + u(x)v(x)w'(x)In words:
  • Differentiate each function one at a time, while keeping the other two functions unchanged.
  • Add all the resulting terms together.
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Step-by-Step Explanation

  1. Start with the product u(x)v(x)w(x)u(x)v(x)w(x). Imagine this as a combination of three "pieces" multiplying together.
  2. Differentiate one function at a time:
    • First, take the derivative of u(x)u(x) (denoted u′(x)u'(x)) and leave v(x)v(x) and w(x)w(x) unchanged.
    • Second, take the derivative of v(x)v(x) (denoted v′(x)v'(x)) and leave u(x)u(x) and w(x)w(x) unchanged.
    • Third, take the derivative of w(x)w(x) (denoted w′(x)w'(x)) and leave u(x)u(x) and v(x)v(x) unchanged.
  3. Combine the results: Add all the terms obtained from the above steps.

Example: Differentiating a Product of 3 Terms

Let’s find the derivative of f(x)=x2⋅sin⁡(x)⋅exf(x) = x^2 \cdot \sin(x) \cdot e^x.
  1. Label the three functions:u(x)=x2,v(x)=sin⁡(x),w(x)=exu(x) = x^2, \quad v(x) = \sin(x), \quad w(x) = e^x
  2. Apply the product rule: Differentiate each function while leaving the others untouched, then sum the results:f′(x)=u′(x)v(x)w(x)+u(x)v′(x)w(x)+u(x)v(x)w′(x)f'(x) = u'(x)v(x)w(x) + u(x)v'(x)w(x) + u(x)v(x)w'(x)
  3. Compute each term:
    • u′(x)=2xu'(x) = 2x
    • v′(x)=cos⁡(x)v'(x) = \cos(x)
    • w′(x)=exw'(x) = e^x
    Substituting into the formula: f′(x)=(2x)(sin⁡(x))(ex)+(x2)(cos⁡(x))(ex)+(x2)(sin⁡(x))(ex)f'(x) = (2x)(\sin(x))(e^x) + (x^2)(\cos(x))(e^x) + (x^2)(\sin(x))(e^x)
  4. Simplify:f′(x)=2xsin⁡(x)ex+x2cos⁡(x)ex+x2sin⁡(x)exf'(x) = 2x \sin(x) e^x + x^2 \cos(x) e^x + x^2 \sin(x) e^x

Key Takeaways

  • The product rule for three terms requires differentiating each function while leaving the others constant.
  • It follows the same additive structure as the two-term product rule.
  • The formula ensures that every term contributes equally to the derivative.
By understanding this extension of the product rule, you can efficiently handle derivatives of products involving multiple terms.
 

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