Video Transcript
Find the first derivative of the function 𝑦 is equal to negative two times the sin of nine 𝑥 minus four multiplied by the cos of nine 𝑥 minus four.
In this question, we’re asked to find the first derivative of a function given in terms of the variable 𝑥. This means we need to differentiate our function with respect to 𝑥. And we can do this in two different ways. We could start by noticing that our function contains the product of two functions with respect to 𝑥, the sin of nine 𝑥 minus four and the cos of nine 𝑥 minus four. Both of these are differentiable functions and we know how to differentiate them. So we could differentiate this by using the product rule.
However, since each of our products is the composition of two functions, a trigonometric function and a linear function, this means we would also need to use the chain rule to differentiate each of our factors. And this is the first method we could use to evaluate this derivative. However, there’s actually an easier method. We can notice the argument of both of these functions is the same. And this can remind us of a useful identity, the double angle formula for sin. The sin of two 𝜃 is equal to two sin 𝜃 multiplied by cos 𝜃. We can then simplify this function by using the double angle formula for sine. We’ll set our value of 𝜃 equal to nine 𝑥 minus four.
We then have two 𝜃 is two multiplied by nine 𝑥 minus four, which, if we distribute the two over our parentheses, is 18𝑥 minus eight. Substituting the expressions for 𝜃 and two 𝜃 into our double angle formula, we have the sin of 18𝑥 minus eight is equal to two sin of nine 𝑥 minus four multiplied by the cos of nine 𝑥 minus four. And this is the negative of the function we’re asked to differentiate. In other words, by using the double angle formula for sine, we’ve shown that 𝑦 is equal to negative the sin of 18𝑥 minus eight. And now our function is just the composition of two differentiable functions.
So we can differentiate this by using the chain rule. And we recall the chain rule tells us the derivative of 𝑓 of 𝑥 composed of 𝑔 of 𝑥 is equal to 𝑔 prime of 𝑥 multiplied by 𝑓 prime evaluated at 𝑔 of 𝑥. And this holds provided 𝑔 of 𝑥 is differentiable at 𝑥 and 𝑓 of 𝑥 is differentiable at 𝑔 of 𝑥. Therefore, to apply the chain rule, we’ll set our inner function 𝑔 of 𝑥 to be 18𝑥 minus eight. And we’ll set our outer function 𝑓 to be negative the sine function. And in the chain rule, we’re using the substitution 𝑢 is equal to 18𝑥 minus eight. In fact, we could call this anything. We could even call this 𝑥.
However, for clarity, we’ll give this a different label 𝑢. To apply the chain rule, we need to differentiate both of these functions, 𝑔 and 𝑓. Let’s start with 𝑔 of 𝑥. 𝑔 of 𝑥 is a linear function. So we’ll differentiate this term by term by using the power rule for differentiation. The derivative of a linear function is just the coefficient of 𝑥. 𝑔 prime of 𝑥 is 18. We then need to differentiate our function 𝑓. And we can do this by recalling the derivative of the sine function is equal to the cosine function. So the derivative of negative sin of 𝑢 with respect to 𝑢 is negative cos of 𝑢.
We can now substitute our functions 𝑔 prime of 𝑥 and 𝑓 prime of 𝑢 and 𝑔 of 𝑥 into our chain rule to evaluate the derivative of 𝑦 with respect to 𝑥. We have the derivative of 𝑦 with respect to 𝑥 is equal to 𝑔 prime of 𝑥 multiplied by 𝑓 prime evaluated at 𝑔 of 𝑥. We have that 𝑔 prime of 𝑥 is equal to 18. We have 𝑓 prime is negative cos of 𝑢. And we need to evaluate this at 𝑔 of 𝑥, which is 18𝑥 minus eight. Therefore, 𝑦 prime is equal to 18 multiplied by negative the cos of 18𝑥 minus eight. And then distributing 18 over our parentheses, we get our final answer. The derivative of 𝑦 is equal to negative two times the sin of nine 𝑥 minus four multiplied by the cos of nine 𝑥 minus four with respect to 𝑥 is equal to negative 18 multiplied by the cos of 18𝑥 minus eight.
