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Video: Differentiating a Combination of Trigonometric and Linear Functions

Chris O’Reilly

Given that 𝑦 = 10𝑥 − 2 cos 9𝑥, determine 𝑑𝑦/𝑑𝑥.

02:18

Video Transcript

Given that 𝑦 equals 10𝑥 minus two cos nine 𝑥, determine 𝑑𝑦 𝑑𝑥.

Well in order to solve this problem, we can actually differentiate each part separately. However, it’s the second part of our function that we’re gonna have to have a look at and see if there’s some general rules that can help us. Well first of all, we can say that if 𝑦 equals cos 𝑥, well we know that 𝑑𝑦 𝑑𝑥 is gonna be equal to negative sin 𝑥.

Okay, so this is useful because actually it helps us differentiate cos 𝑥. However, the term that we’re looking in is in a slightly different form, because our term is actually more in this form which is 𝑦 is equal to 𝑎 cos 𝑓 𝑥.

So what it actually means is a constant, so in our case which would be negative two, and then multiplied by cosine of our function itself. Well if we have it in this form, we can say that 𝑑𝑦 𝑑𝑥 is gonna be equal to negative 𝑎 multiplied by the derivative of our function multiplied by the sine of our function.

Okay, great. So now we’ve got this general term. We can use this to help us actually differentiate and determine 𝑑𝑦 𝑑𝑥. So we’ve got 𝑦 is equal to 10𝑥 minus two cos nine 𝑥. So we’re gonna get 𝑑𝑦 𝑑𝑥 is equal to 10. We get 10 because if we differentiate the first term, it gives us 10.

Well just to remind us how we do that, how to differentiate, what we do is we actually multiply the coefficient by the exponent, so 10 by one which gives us 10, and then we reduce the exponent by one. So we had 𝑥 power of one. Well one minus one is zero. But we know that anything to the power of zero is just one, so we get 10𝑥 to the power of zero which is just equal to 10.

Okay, great. So we can get on to the rest of our differentiation. So then we’re gonna have minus negative two multiplied by the derivative with respect to 𝑥 of nine 𝑥 multiplied by sin nine 𝑥, which is gonna be equal to 10 minus negative two multiplied by nine because actually if we differentiate with respect to 𝑥 nine 𝑥, we just get nine multiplied by sin nine 𝑥, which is gonna be equal to 10 minus negative 18 sin nine 𝑥.

So therefore, we can say that given that 𝑦 equals 10𝑥 minus two cos nine 𝑥, 𝑑𝑦 𝑑𝑥 is gonna be equal to 10 plus 18 sin nine 𝑥.