Video: Differentiating Trigonometric Functions

Find 𝑑𝑦/𝑑π‘₯, given that 𝑦 = 6 sin 3π‘₯.

02:21

Video Transcript

Find 𝑑𝑦 by 𝑑π‘₯, given that 𝑦 equals six sin three π‘₯.

We are looking then to differentiate six sin three π‘₯ with respect to π‘₯. And to find this, we’ll need to use the fact that the derivative of sin π‘₯ with respect to π‘₯ is cos π‘₯. This is with π‘₯ in radians. Using the fact that the derivative of a number times a function is that number times the derivative of the function, we can see that 𝑑 by 𝑑π‘₯ of six sin π‘₯ is six cos π‘₯. But we’re looking for the derivative of six sin three π‘₯. How do we find this? Well, we have to use the chain rule.

To make it easier to apply the chain rule, we will define a new variable 𝑧 to be three π‘₯. Then, as 𝑦 is equal to six sin three π‘₯ as we’re told in the question, 𝑦 is equal to six sin 𝑧. Now, how does this help? Well, the chain rule tells us that the derivative of 𝑦 with respect to π‘₯ is the derivative of 𝑦 with respect to 𝑧 times the derivative of 𝑧 with respect to π‘₯.

Let’s apply this. We need to find 𝑑𝑦 by 𝑑𝑧. And so we use the expression for 𝑦 in terms of 𝑧: 𝑦 equals six sin 𝑧. And just as the derivative with respect to π‘₯ of six sin π‘₯ is six cos π‘₯, the derivative with respect to 𝑧 of six sin 𝑧 is six cos 𝑧.

Now, we just need to find 𝑑𝑧 by 𝑑π‘₯. And as 𝑧 equals three π‘₯, 𝑑𝑧 by 𝑑π‘₯ is 𝑑 by 𝑑π‘₯ over three π‘₯, which is three. So altogether, 𝑑𝑦 by 𝑑π‘₯ is six cos 𝑧 times three which is 18 cos 𝑧. And we don’t want 𝑑𝑦 by 𝑑π‘₯ written in terms of some other variable 𝑧. We’d like it written in terms of π‘₯ if possible. Using the fact that 𝑧 is three π‘₯, we see that 𝑑𝑦 by 𝑑π‘₯ is 18 cos three π‘₯. And this is our final answer.

If we know that the derivative of sin π‘₯ with respect to π‘₯ is cos π‘₯, then we can find the derivative of lots of expressions involving sin π‘₯ by using the chain rule.

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