Video: EG19M1-DiffAndInt-Q13

If 𝑦 = log |sin π‘₯|, find d𝑦/dπ‘₯.

02:21

Video Transcript

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

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 π‘₯, we take the derivative of 𝑓 with respect to 𝑔 of π‘₯ and then multiply that by the derivative of what’s inside the derivative of 𝑔 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‘Ž equals one over the natural log of π‘Ž times π‘₯. In this case, our log base hasn’t been specified. So we can just leave it as π‘Ž. We’ll have one over sin of π‘₯ times the natural log of π‘Ž, which we’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‘Ž 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‘Ž, like this, the natural log of 𝑒 over the natural log of π‘Ž. 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 π‘Ž equals the log of 𝑒 base π‘Ž.

The derivative of 𝑦 equals log of sin π‘₯ equals cot of π‘₯ times the log of 𝑒.

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