If 𝑦 equals the log of sin of 𝑥, find the derivative of 𝑦 with respect to 𝑥.
Almost immediately, we should notice that we’re taking a function of a function. And to take the derivative of something in the format 𝑓of 𝑔 of 𝑥, we take the derivative of 𝑓 with respect to 𝑔 of 𝑥 and then multiply that by the derivative of what’s inside the derivative of 𝑔 of 𝑥. We need to take the derivative of the log of sin of 𝑥 and multiply that by the derivative of sin of 𝑥. We know that the derivative of sin of 𝑥 is cos of 𝑥. So we can substitute that in.
We also need to know another form. We need to know derivative of log of 𝑥, with base 𝑎 equals one over the natural log of 𝑎 times 𝑥. In this case, our log base hasn’t been specified. So we can just leave it as 𝑎. We’ll have one over sin of 𝑥 times the natural log of 𝑎, which we’re multiplying by cos of 𝑥. We can break up the factors like this, one over sin 𝑥 times one over the natural log of 𝑎 times cos of 𝑥.
If we multiply cos times one over sin, we get cos of 𝑥 over sin of 𝑥. And the ratio of cos of 𝑥 over sin of 𝑥 equals cot of 𝑥. We can also rewrite one over the natural log of 𝑎, like this, the natural log of 𝑒 over the natural log of 𝑎. The natural log of 𝑒 equals one. So we haven’t changed the value of this fraction. We write it in that format because then we can simplify this fraction. The natural log of 𝑒 over the natural log of 𝑎 equals the log of 𝑒 base 𝑎.
The derivative of 𝑦 equals log of sin 𝑥 equals cot of 𝑥 times the log of 𝑒.