# Video: EG19M1-DiffAndInt-Q13

If π¦ = log |sin π₯|, find dπ¦/dπ₯.

02:21

### Video Transcript

If π¦ equals the log of sin of π₯, find the derivative of π¦ with respect to π₯.

Almost immediately, we should notice that weβre taking a function of a function. And to take the derivative of something in the format πof π of π₯, we take the derivative of π with respect to π of π₯ and then multiply that by the derivative of whatβs inside the derivative of π of π₯. We need to take the derivative of the log of sin of π₯ and multiply that by the derivative of sin of π₯. We know that the derivative of sin of π₯ is cos of π₯. So we can substitute that in.

We also need to know another form. We need to know derivative of log of π₯, with base π equals one over the natural log of π times π₯. In this case, our log base hasnβt been specified. So we can just leave it as π. Weβll have one over sin of π₯ times the natural log of π, which weβre multiplying by cos of π₯. We can break up the factors like this, one over sin π₯ times one over the natural log of π times cos of π₯.

If we multiply cos times one over sin, we get cos of π₯ over sin of π₯. And the ratio of cos of π₯ over sin of π₯ equals cot of π₯. We can also rewrite one over the natural log of π, like this, the natural log of π over the natural log of π. The natural log of π equals one. So we havenβt changed the value of this fraction. We write it in that format because then we can simplify this fraction. The natural log of π over the natural log of π equals the log of π base π.

The derivative of π¦ equals log of sin π₯ equals cot of π₯ times the log of π.