Video: Differentiating Trigonometric Functions Using the Chain Rule

If 𝑦 = cos (8/𝑥⁵), find d𝑦/d𝑥.

02:22

Video Transcript

If 𝑦 equals cos of eight over 𝑥 to the power of five, find d𝑦 by d𝑥.

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 d𝑥 is equal to d𝑦 by d𝑢 multiplied by d𝑢 by d𝑥. Our first step is to let 𝑢 equal eight over 𝑥 to the power of five. This can be rewritten as eight 𝑥 to the power of negative five. Differentiating this with respect to 𝑥 gives us negative 40𝑥 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 𝑥 to the power of six.

Therefore, d𝑢 by d𝑥 is equal to negative 40 over 𝑥 to the power of six. Substituting in 𝑢 to our initial equation gives us 𝑦 is equal to cos 𝑢. We know that the derivative of cos 𝑥 is negative sin 𝑥. This means that differentiating 𝑦 equals cos 𝑢 with respect to 𝑢 gives us d𝑦 by d𝑢 is equal to negative sin 𝑢.

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

d𝑦 by d𝑥 is, therefore, equal to 40 over 𝑥 to the power of six sin eight over 𝑥 to the power of five. This is the derivative of cos eight over 𝑥 to the power of five.

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