Question Video: Evaluating an Expression Containing the Derivative of a Function Using the Product Rule Mathematics

Video Transcript

Given that 𝑦 is equal to 𝑥 cubed plus 𝑥 squared plus eight 𝑥 and 𝑧 is equal to 𝑥 multiplied by 𝑥 minus four multiplied by 𝑥 minus one, determine 𝑑𝑦 𝑑𝑥 minus 𝑑𝑧 𝑑𝑥.

Let’s first consider 𝑦 equals 𝑥 cubed plus 𝑥 squared plus eight 𝑥. In order to calculate 𝑑𝑦 𝑑𝑥, we need to differentiate this equation. Differentiating 𝑥 cubed gives us three 𝑥 squared, differentiating 𝑥 squared gives us two 𝑥, and differentiating eight 𝑥 gives us eight. Therefore, 𝑑𝑦 by 𝑑𝑥 is equal to three 𝑥 squared plus two 𝑥 plus eight.

Now let’s consider our second equation: 𝑧 is equal to 𝑥 multiplied by 𝑥 minus four multiplied by 𝑥 minus one. Before we differentiate this equation, we need to expand and simplify. Expanding the two parentheses gives us 𝑥 squared minus 𝑥 minus four 𝑥 plus four.

We now need to multiply all of these terms by 𝑥. Multiplying the new parenthesis by 𝑥 gives us 𝑥 cubed minus 𝑥 squared minus four 𝑥 squared plus four 𝑥. Simplifying by grouping like terms gives us that 𝑧 is equal to 𝑥 cubed minus five 𝑥 squared plus four 𝑥.

Differentiating this gives us 𝑑𝑧 𝑑𝑥 equals three 𝑥 squared minus 10𝑥 plus four as the differential of 𝑥 cubed is three 𝑥 squared, the differential of negative five 𝑥 squared is negative 10𝑥, and the differential of four 𝑥 is equal to four.

We now have two expressions 𝑑𝑦 by 𝑑𝑥 is equal to three 𝑥 squared plus two 𝑥 plus eight and 𝑑𝑧 𝑑𝑥 is equal to three 𝑥 squared minus 10𝑥 plus four. In order to calculate 𝑑𝑦 by 𝑑𝑥 minus 𝑑𝑧 by 𝑑𝑥, we need to subtract these two expressions. Three 𝑥 squared minus three 𝑥 squared is equal to zero, positive two 𝑥 minus negative 10𝑥 is equal to 12𝑥, and positive eight minus positive four is equal to four.

This means that three 𝑥 squared plus two 𝑥 plus eight minus three 𝑥 squared minus 10𝑥 plus four is equal to 12𝑥 plus four. The value of the 𝑑𝑦 𝑑𝑥 minus 𝑑𝑧 𝑑𝑥 is equal to 12 𝑥 plus four.