Understanding the Product Rule With 3 Terms - The Product Rule is a concept used in calculus to help solve problems with polynomials of more than two terms. In some cases, the polynomial can have three terms. The Product Rule with three terms states that the product of a derivative and a function must be differentiated. It is an important concept to understand when learning calculus

Here’s a detailed explanation of the Product Rule with 3 Terms, broken down for clarity.
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)dxd[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)u(x)v(x)w(x).The Product Rule for 3 Terms
For three functions u(x)u(x)u(x), v(x)v(x)v(x), and w(x)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)dxd[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.
Step-by-Step Explanation
- Start with the product u(x)v(x)w(x)u(x)v(x)w(x)u(x)v(x)w(x). Imagine this as a combination of three "pieces" multiplying together.
- Differentiate one function at a time:
- First, take the derivative of u(x)u(x)u(x) (denoted u′(x)u'(x)u′(x)) and leave v(x)v(x)v(x) and w(x)w(x)w(x) unchanged.
- Second, take the derivative of v(x)v(x)v(x) (denoted v′(x)v'(x)v′(x)) and leave u(x)u(x)u(x) and w(x)w(x)w(x) unchanged.
- Third, take the derivative of w(x)w(x)w(x) (denoted w′(x)w'(x)w′(x)) and leave u(x)u(x)u(x) and v(x)v(x)v(x) unchanged.
- 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^xf(x)=x2⋅sin(x)⋅ex.- 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^xu(x)=x2,v(x)=sin(x),w(x)=ex
- 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)f′(x)=u′(x)v(x)w(x)+u(x)v′(x)w(x)+u(x)v(x)w′(x)
- Compute each term:
- u′(x)=2xu'(x) = 2xu′(x)=2x
- v′(x)=cos(x)v'(x) = \cos(x)v′(x)=cos(x)
- w′(x)=exw'(x) = e^xw′(x)=ex
- 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^xf′(x)=2xsin(x)ex+x2cos(x)ex+x2sin(x)ex
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.