Video Transcript
Find 𝑑𝑦 by 𝑑𝑥, given that 𝑦 is equal to seven 𝑥 to the power of five plus one divided by 𝑥 to the power of six.
One divided by 𝑥 to the power of 𝑛 is equal to 𝑥 to the power of negative 𝑛. This means that we can rewrite one divided by 𝑥 to the power of six as 𝑥 to the power of negative six. If 𝑦 is equal to 𝑎 multiplied by 𝑥 to the power of 𝑛, then the differential 𝑑𝑦 by 𝑑𝑥 is equal to 𝑛 multiplied by 𝑎 multiplied by 𝑥 to the power of 𝑛 minus one.
Differentiating the first term seven 𝑥 to the power of five gives us 35𝑥 to the power of four, as five multiplied by seven is 35 and reducing the power by one gives us 𝑥 to the power of four. Differentiating the second term gives us negative six 𝑥 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 𝑥 to the power of seven.
This means that if 𝑦 is equal to seven 𝑥 to the power of five plus one divided by 𝑥 to the power of six, then 𝑑𝑦 by 𝑑𝑥 is equal to 35𝑥 to the power of four minus six divided by 𝑥 to the power of seven.
