- #1
DB
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simple question, if the derivative of cos(x) as x->0 is 1, the is the derivative of cos(3x) as x->0 = 1 aswell?
Derivative of cos(3x) at x = 0: Is it 1?
- Thread starter DB
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In summary, the derivative of cos(x) is -sin(x) and the derivative of cos(3x) is -3sin(3x). The derivative of a function of x can approach a limit as well, and in this case, the derivative of cos(3x) as x->0 will approach 0, not 1. The chain rule can be applied to find the derivative of cos(3x) by substituting u=3x and applying the chain rule.
- #2
AKG
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The derivative of cos(x) as x->0 is -sin(x) as x->0 is -sin(0) is 0, not 1. But simply cos(x) as x->0 is cos(0) = 1. the derivative of cos(3x) as x->0 = -3sin(3x) as x->0 = -3sin(3*0) = -3sin(0) = -3*0 = 0. cos(3x) as x->0 = cos(3*0) = cos(0) = 1. The derivative of cos(x) as x->a = -sin(x) as x->a = -sin(a). The derivative of cos(3x) as x->a = -3sin(3x) as x->a = -3sin(3a).
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- #3
mathman
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The derivative of cos(x) is -sin(x).
- #4
AKG
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Yeah, I knew that. 
Edited.
Edited.- #5
moose
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Think about what a derivative means. This should be plenty enough to give you the answer you're looking for.
- #6
HallsofIvy
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DB said:
The derivative of sin x goes to 1 as x goes to 0. No, the derivative of sin(3x) does not go to 1. It goes to 3. Do you see why?
- #7
cDimino
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Curious... it looks like you're looking for a limit. I wasn't aware that derivitaves "approached" anything.
- #8
HallsofIvy
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The derivative of a function of x is a function of x and can approach a limit as well as any function.cDimino said:
- #9
moose
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The derivative tells you how the result will change with a change in the x. If it were to linearly change, the derivative would say how y changes when 1 is added to x. So, if you have cos(3x), the derivative will have to be 3 times greater than cosx, because if you raise x by an amount (say, 1), you are raising the quantity inside the cos by 3 times that amount. It's the same concept as the derivative (slope) of y=x and y=3x.
This is how the chain rule can be derived...
This is how the chain rule can be derived...
- #10
Robokapp
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just U-substitute u=3x if you must and apply chain rule. dy/dx=dy/du*du/dx
FAQ: Derivative of cos(3x) at x = 0: Is it 1?
What is the derivative of cos(3x) at x = 0?
The derivative of cos(3x) at x = 0 is -3sin(3x).
How do you calculate the derivative of cos(3x) at x = 0?
The derivative of cos(3x) at x = 0 can be calculated using the chain rule, which states that the derivative of f(g(x)) is equal to f'(g(x)) * g'(x). In this case, f(x) = cos(x) and g(x) = 3x. Therefore, the derivative is equal to -3sin(3x).
Why is the derivative of cos(3x) at x = 0 equal to -3sin(3x)?
The derivative of cos(3x) is equal to -3sin(3x) because the derivative of cos(x) is equal to -sin(x), and the derivative of 3x is equal to 3. When these two functions are combined using the chain rule, the result is -3sin(3x).
Can you prove that the derivative of cos(3x) at x = 0 is -3sin(3x)?
Yes, the derivative of cos(3x) at x = 0 can be proven using the definition of the derivative, which states that the derivative of a function f(x) at a point x = a is equal to the limit as h approaches 0 of [f(a + h) - f(a)] / h. By plugging in x = 0 and evaluating the limit, the result is -3sin(3x).
Is the derivative of cos(3x) at x = 0 always equal to -3sin(3x)?
Yes, the derivative of cos(3x) at x = 0 is always equal to -3sin(3x). This is because the derivative of cosine is always equal to negative sine, and the coefficient of x (in this case, 3) remains the same when taking the derivative. Therefore, the derivative will always be -3sin(3x).