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

Question Video: Evaluating an Expression Containing the Derivative of a Function Using the Product Rule Mathematics • Second Year of Secondary School

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Given that π¦ = π₯Β³ + π₯Β² + 8π₯ and π§ = π₯(π₯ β 4)(π₯ β 1), determine ππ¦/ππ₯ β ππ§/ππ₯.

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Video Transcript

Given that π¦ is equal to π₯ cubed plus π₯ squared plus eight π₯ and π§ is equal to π₯ multiplied by π₯ minus four multiplied by π₯ minus one, determine ππ¦ ππ₯ minus ππ§ ππ₯.

Letβs first consider π¦ equals π₯ cubed plus π₯ squared plus eight π₯. In order to calculate ππ¦ ππ₯, 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, ππ¦ by ππ₯ is equal to three π₯ squared plus two π₯ plus eight.

Now letβs consider our second equation: π§ 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 π§ is equal to π₯ cubed minus five π₯ squared plus four π₯.

Differentiating this gives us ππ§ ππ₯ 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 ππ¦ by ππ₯ is equal to three π₯ squared plus two π₯ plus eight and ππ§ ππ₯ is equal to three π₯ squared minus 10π₯ plus four. In order to calculate ππ¦ by ππ₯ minus ππ§ by ππ₯, 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 ππ¦ ππ₯ minus ππ§ ππ₯ is equal to 12 π₯ plus four.

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