Question Video: Differentiating Trigonometric Functions Using the Chain Rule Mathematics • Higher Education

If 𝑦 = cos (8/π‘₯⁡), find d𝑦/dπ‘₯.

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Video Transcript

If 𝑦 equals cos of eight over π‘₯ to the power of five, find d𝑦 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 d𝑦 by dπ‘₯ is equal to d𝑦 by d𝑒 multiplied by d𝑒 by dπ‘₯. Our first step is to let 𝑒 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, d𝑒 by dπ‘₯ is equal to negative 40 over π‘₯ to the power of six. Substituting in 𝑒 to our initial equation gives us 𝑦 is equal to cos 𝑒. We know that the derivative of cos π‘₯ is negative sin π‘₯. This means that differentiating 𝑦 equals cos 𝑒 with respect to 𝑒 gives us d𝑦 by d𝑒 is equal to negative sin 𝑒.

We now have values for d𝑦 by d𝑒 and d𝑒 by dπ‘₯. d𝑦 by dπ‘₯ will be equal to the product of these two terms. As we need our answer in terms of π‘₯, 𝑒 can be replaced by eight over π‘₯ to the power of five. d𝑦 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.

d𝑦 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.

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