Video: Finding the First Derivative of a Function Using the Power Rule

Find 𝑑𝑦/𝑑π‘₯, given that 𝑦 = 7π‘₯⁡ + (1/π‘₯⁢).

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Video Transcript

Find 𝑑𝑦 by 𝑑π‘₯, given that 𝑦 is equal to seven π‘₯ to the power of five plus one divided by π‘₯ to the power of six.

One divided by π‘₯ to the power of 𝑛 is equal to π‘₯ to the power of negative 𝑛. This means that we can rewrite one divided by π‘₯ to the power of six as π‘₯ to the power of negative six. If 𝑦 is equal to π‘Ž multiplied by π‘₯ to the power of 𝑛, then the differential 𝑑𝑦 by 𝑑π‘₯ is equal to 𝑛 multiplied by π‘Ž multiplied by π‘₯ to the power of 𝑛 minus one.

Differentiating the first term seven π‘₯ to the power of five gives us 35π‘₯ to the power of four, as five multiplied by seven is 35 and reducing the power by one gives us π‘₯ to the power of four. Differentiating the second term gives us negative six π‘₯ to the power of negative seven, as negative six minus one is negative seven. The second term can be rewritten as negative six divided by π‘₯ to the power of seven.

This means that if 𝑦 is equal to seven π‘₯ to the power of five plus one divided by π‘₯ to the power of six, then 𝑑𝑦 by 𝑑π‘₯ is equal to 35π‘₯ to the power of four minus six divided by π‘₯ to the power of seven.

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