Video: Differentiating Trigonometric Functions Using the Chain Rule

Video Transcript

If 𝑦 equals negative eight sin of sin of six 𝑥 minus cos of sin of six 𝑥, find d𝑦 by d𝑥.

We can differentiate this term-by-term. However, each of our terms is a function of a function. And when we want to differentiate a function of a function, we use the chain rule. Let’s remind ourselves of the chain rule. This says that if 𝑦 equals 𝑓 of 𝑢 and 𝑢 equals 𝑔 of 𝑥, then d𝑦 by d𝑥 equals d𝑦 by d𝑢 multiplied by d𝑢 by d𝑥.

Let’s call this first term 𝑧. And to avoid confusion with the 𝑦 in the question and the 𝑦 in the chain rule formula, let’s replace 𝑦 in the formula with 𝑧. We let our inner function 𝑢 equal sin of six 𝑥. And so, 𝑧 equals negative eight sin of 𝑢. To use the chain rule formula, we need d𝑧 by d𝑢 and d𝑢 by d𝑥. Let’s start by finding d𝑧 by d𝑢.

This is going to be the derivative of negative eight sin of 𝑢 with respect to 𝑢. To do this, we remember the main derivatives of trigonometric functions. And we see that the derivative of sin of 𝑥 is cos of 𝑥. And so, the derivative of negative eight sin of 𝑢 with respect to 𝑢 is negative eight cos of 𝑢. And now, we find d𝑢 by d𝑥. This is the derivative of sin of six 𝑥 with respect to 𝑥.

If we use the fact that the derivative of sin of 𝑎𝑥 equals 𝑎 cos of 𝑎𝑥, then d𝑢 by d𝑥 equals six cos of six 𝑥. And now, we can apply the formula for the chain rule. Negative eight sin of sin of six 𝑥 differentiates to negative eight cos of 𝑢 multiplied by six cos of six 𝑥. We remember that we let 𝑢 equal sin of six 𝑥. And so, we can replace 𝑢 with sin of six 𝑥.

And now, we move on to our next term. Let’s call this function ℎ. And so, we replace 𝑧 in our formula for the chain rule with ℎ. And we’ll replace 𝑢 with 𝑤 because we already assigned 𝑢 in our previous term. So, 𝑤 is our inner, function sin of six 𝑥, which means that ℎ equals cos of 𝑤. We need to find dℎ by d𝑤 and d𝑤 by d𝑥. Let’s start by finding dℎ by d𝑤.

This is the derivative of cos of 𝑤 with respect to 𝑤. We can use the fact that the derivative of cos of 𝑥 with respect to 𝑥 is negative sin of 𝑥. So, dℎ by d𝑤 equals negative sin of 𝑤. We also need to find d𝑤 by d𝑥. This is the derivative of sin of six 𝑥 with respect to 𝑥. Using the fact that the derivative of sin of 𝑎𝑥 with respect to 𝑥 is 𝑎 cos of 𝑎𝑥, d𝑤 by d𝑥 equals six cos of six 𝑥. And so, cos of sin of six 𝑥 differentiates to negative sin of 𝑤 multiplied by six cos of six 𝑥.

We remember that we let 𝑤 equal sin of six 𝑥. And so, we replace 𝑤 with sign of six 𝑥. And now, we’ve seen what each term differentiates to, we can put it together to get d𝑦 by d𝑥. d𝑦 by d𝑥 equals negative eight cos of sin of six 𝑥 multiplied by six cos of six 𝑥 minus, because it was a minus in the question, negative sin of sin of six 𝑥 multiplied by six cos of six 𝑥.

Here, we are subtracting a negative, so we can just write this as an add. And six cos of six 𝑥 appears in both terms, so we can take this out as a common factor. This leaves us with six cos of six 𝑥 all multiplied by negative eight cos of sin of six 𝑥 add sin of sin of six 𝑥.