## \dfrac{d}{dx}\sin(Y) \neq \cos(Y)##Hawksteinman said:I was thinking and came up with this. I know it's wrong but can't find the mistake :(
dy/dx sin(x) = cos(x)
dy/dx sin(kx) = kcos(kx)
So dy/dx sin(3x) = 3cos(3x)
Now let Y = 3x
dy/dx sin(Y) = cos(Y) = cos(3x)
3cos(3x) = cos(3x)
3 = 1
Where is the mistake?
mfb said:Your notation doesn’t make sense.
dy/dx is the derivative of y with respect to x. “dy/dx sin(x)” is not a well-defined expression.
What you nean is d/dx sin(x). And suddenly the issue disappears:
d/dx sin(3x)=3cos(x)
d/dx sin(y)=? - here we need the chain rule and d/dx 3x = 3:
d/dx sin(y) = cos(y) d/dx y = cos(y) * 3
As already noted, the above should be ##\frac d {dx}\left(\sin(3x)\right) = 3\cos(3x)##Hawksteinman said:So dy/dx sin(3x) = 3cos(3x)
Actually, you used the chain rule in the first line of what I quoted, above. The chain rule is what gives you that leading factor of 3.Hawksteinman said:I haven't done the chain rule yet I'll need to look into that :)
Mark44 said:As already noted, the above should be ##\frac d {dx}\left(\sin(3x)\right) = 3\cos(3x)##
##\frac{dy}{dx}## is the derivative of y with respect to x, so it is a thing, a noun.
##\frac d{dx}## is the operator that signifies taking the derivative of something with respect to x. It is an action that hasn't completed yet, a verb. Don't confuse these two things.
Actually, you used the chain rule in the first line of what I quoted, above. The chain rule is what gives you that leading factor of 3.
They were using the chain rule in the table.Hawksteinman said:I don't know, I just used a table of derivatives :/
The sin function, also known as the sine function, is a mathematical function that represents the relationship between the angles and sides of a right triangle. It is defined as the ratio of the opposite side to the hypotenuse of a right triangle.
Differentiation is a mathematical process that involves finding the rate of change of a function with respect to its independent variable. It is a fundamental concept in calculus and is used to analyze how a function changes over time or space.
To differentiate a sin function, we use the basic differentiation rules of calculus. The derivative of sinx is cosx, which means that the value of the derivative at any point on the sin function will be equal to the value of the cosine function at that point.
Common mistakes when differentiating a sin function include forgetting to use the chain rule, not applying the power rule correctly, and mixing up the signs in the final answer. It is important to carefully follow the rules of differentiation and double-check your work to avoid these mistakes.
To check if you have made a mistake while differentiating a sin function, you can use the second derivative test. This involves finding the derivative of the derivative and plugging in values to see if they match the original function. If they do not match, it is likely that you have made a mistake in your differentiation process.