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

Given that 𝑦 = π‘₯Β³ + π‘₯Β² + 8π‘₯ and 𝑧 = π‘₯(π‘₯ βˆ’ 4)(π‘₯ βˆ’ 1), determine 𝑑𝑦/𝑑π‘₯ βˆ’ 𝑑𝑧/𝑑π‘₯.

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Video Transcript

Given that 𝑦 is equal to π‘₯ cubed plus π‘₯ squared plus eight π‘₯ and 𝑧 is equal to π‘₯ multiplied by π‘₯ minus four multiplied by π‘₯ minus one, determine 𝑑𝑦 𝑑π‘₯ minus 𝑑𝑧 𝑑π‘₯.

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

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

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

Differentiating this gives us 𝑑𝑧 𝑑π‘₯ equals three π‘₯ squared minus 10π‘₯ plus four as the differential of π‘₯ cubed is three π‘₯ squared, the differential of negative five π‘₯ squared is negative 10π‘₯, and the differential of four π‘₯ is equal to four.

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

This means that three π‘₯ squared plus two π‘₯ plus eight minus three π‘₯ squared minus 10π‘₯ plus four is equal to 12π‘₯ plus four. The value of the 𝑑𝑦 𝑑π‘₯ minus 𝑑𝑧 𝑑π‘₯ is equal to 12 π‘₯ plus four.

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