Video: Differentiating Polynomials

Find 𝑑𝑦/𝑑𝑥, given that 𝑦 = −8𝑥⁹ − 4𝑥⁵ + 3𝑥 + 5.

02:13

Video Transcript

Find 𝑑𝑦 by 𝑑𝑥, given that 𝑦 is equal to negative eight 𝑥 to the power of nine minus four 𝑥 to the power of five plus three 𝑥 plus five.

In order to answer this question, we need to understand the process of differentiation. If 𝑦 is equal to 𝑎 multiplied by 𝑥 to the power of 𝑛, then 𝑑𝑦 by 𝑑𝑥 is equal to 𝑛 multiplied by 𝑎 multiplied by 𝑥 to the power of 𝑛 minus one.

We multiply the power by the coefficient and then subtract one from the power. In our case, 𝑦 is equal to negative eight 𝑥 to the power of nine minus four 𝑥 to the power of five plus three 𝑥 plus five. In order to find 𝑑𝑦 by 𝑑𝑥, we’ll differentiate each term individually.

The differential of negative eight 𝑥 to the power of nine is negative 72 𝑥 to the power of eight as nine multiplied by negative eight is negative 72. Differentiating negative four 𝑥 to the power of five gives us negative 20 𝑥 to the power of four as five multiplied by negative four is negative 20 and we decrease the power by one.

Differentiating three 𝑥 gives us three as one multiplied by three is equal to three and subtracting one from the power gives us 𝑥 to the power of zero. This is equal to one. Therefore, the differential of three 𝑥 is equal to three. Differentiating five gives us zero. Any number that we differentiate will always give us an answer of zero.

𝑑𝑦 by 𝑑𝑥 is, therefore, equal to negative 72 𝑥 to the power of eight minus 20 𝑥 to the power of four plus three.

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