### Video Transcript

If π¦ equals cos of eight over π₯ to the power of five, find dπ¦ by dπ₯.

To solve this problem involving differentiation, we will use the chain rule as this is a composite function, otherwise known as a function of a function. The chain rule states that dπ¦ by dπ₯ is equal to dπ¦ by dπ’ multiplied by dπ’ by dπ₯. Our first step is to let π’ equal eight over π₯ to the power of five. This can be rewritten as eight π₯ to the power of negative five. Differentiating this with respect to π₯ gives us negative 40π₯ to the power of negative six. We multiply the negative five by eight and decrease the power by one. This can be rewritten as negative 40 over π₯ to the power of six.

Therefore, dπ’ by dπ₯ is equal to negative 40 over π₯ to the power of six. Substituting in π’ to our initial equation gives us π¦ is equal to cos π’. We know that the derivative of cos π₯ is negative sin π₯. This means that differentiating π¦ equals cos π’ with respect to π’ gives us dπ¦ by dπ’ is equal to negative sin π’.

We now have values for dπ¦ by dπ’ and dπ’ by dπ₯. dπ¦ by dπ₯ will be equal to the product of these two terms. As we need our answer in terms of π₯, π’ can be replaced by eight over π₯ to the power of five. dπ¦ by dπ₯ is equal to negative 40 over π₯ to the power of six multiplied by negative sin of eight over π₯ to the power of five. Multiplying two negative terms gives us a positive answer.

dπ¦ by dπ₯ is, therefore, equal to 40 over π₯ to the power of six sin eight over π₯ to the power of five. This is the derivative of cos eight over π₯ to the power of five.