Find d𝑦 by d𝑥 if 𝑦 is equal to the tan of five cot 𝑥.
Our function here is a composite function, so we will use the chain rule to differentiate it. This states that d𝑦 by d𝑥 is equal to d𝑦 by d𝑢 multiplied by d𝑢 d𝑥. We begin by letting 𝑢 equal five cot 𝑥. We know that the derivative of cot 𝑥 is negative cosec squared 𝑥. This means that d𝑢 by d𝑥 is equal to negative five cosec squared 𝑥. Replacing five cot 𝑥 with 𝑢 in our initial equation gives us 𝑦 is equal to tan 𝑢.
Once again, we know that the derivative of tan 𝑥 is sec squared 𝑥. If 𝑦 is equal to tan 𝑢, then d𝑦 by d𝑢 is equal to sec squared 𝑢. We now have expressions for d𝑦 by d𝑢 and d𝑢 by d𝑥. In order to work out d𝑦 by d𝑥, both of these need to be in terms of 𝑥. We can, therefore, rewrite d𝑦 by d𝑢 as sec squared of five cot 𝑥. d𝑦 by d𝑥 is, therefore, equal to negative five cosec squared 𝑥 multiplied by sec squared of five cot 𝑥.