If 𝑦 equals 15 sin squared eight 𝑥 minus 15 cos squared eight 𝑥, find d𝑦 by d𝑥.
In order to work out an expression for d𝑦 by d𝑥, we need to differentiate 𝑦 with respect to 𝑥. In this case, we could differentiate each term individually. We could differentiate 15 sin squared eight 𝑥 and negative 15 cos squared eight 𝑥. However, in this case, there is a shortcut using one of the double-angle formulae. Cos two 𝑥 is equal to cos squared 𝑥 minus sin squared 𝑥.
Both of the terms in our original equation have a coefficient of 15. This means that we can factorize 15 out of the equation. Inside the bracket, we’re left with sin squared eight 𝑥 minus cos squared eight 𝑥. If we multiply each of the three terms of the double-angle formula by negative one, we get negative cos two 𝑥 is equal to negative cos squared 𝑥 plus sin squared 𝑥. The right-hand side of this equation can be rewritten as sin squared 𝑥 minus cos squared 𝑥. This is very similar to the expression inside the bracket, sin squared eight 𝑥 minus cos squared eight 𝑥.
Using the double-angle formula, we can see that this would be equal to negative of cos 16𝑥. 𝑦 is equal to 15 multiplied by negative cos 16𝑥. This can be rewritten as negative 15 cos 16𝑥. Our next step is to use one of the rules of differentiating. If 𝑦 is equal to 𝑎 cos 𝑛𝑥, then d𝑦 by d𝑥 is equal to negative 𝑎𝑛 sin 𝑛𝑥.
In order to differentiate negative 15 cos 16𝑥, we firstly need to multiply the negative of negative 15 by 16. And, the cos of 16𝑥 becomes the sin of 16𝑥. The two negative signs will become a positive. So, we need to multiply 15 by 16. 16 multiplied by 10 is equal to 160. 16 multiplied by five is equal to 80. This means that 16 multiplied by 15 is equal to 240, as 160 plus 80 equals 240. d𝑦 by d𝑥 is equal to 240 sin 16𝑥. If 𝑦 is equal to 15 sin squared eight 𝑥 minus 15 cos squared eight 𝑥, then d𝑦 by d𝑥 is 240 sin 16𝑥.