If y=cos x, what is y^(6) (x)?
same as y": -cosx
To find y^(6)(x), we need to differentiate y(x) six times with respect to x.
Given that y = cos x, we can start by taking the first derivative:
y' = -sin x
Then, we take the second derivative:
y'' = -cos x
Next, we take the third derivative:
y''' = sin x
The fourth derivative:
y'''' = cos x
The fifth derivative:
y''''' = -sin x
Finally, we take the sixth derivative:
y^(6) = -cos x
Therefore, y^(6)(x) = -cos x.
To find the expression for y^(6)(x), we need to take the sixth derivative of y = cos(x) with respect to x.
Let's start by finding the first derivative of y = cos(x):
dy/dx = -sin(x)
The second derivative can be found by taking the derivative of (-sin(x)):
d²y/dx² = -cos(x)
The third derivative can be found by taking the derivative of (-cos(x)):
d³y/dx³ = sin(x)
The fourth derivative can be found by taking the derivative of (sin(x)):
d⁴y/dx⁴ = cos(x)
The fifth derivative can be found by taking the derivative of (cos(x)):
d⁵y/dx⁵ = -sin(x)
Finally, we can find the sixth derivative by taking the derivative of (-sin(x)):
d⁶y/dx⁶ = -cos(x)
Therefore, y^(6)(x) = -cos(x).
So, the answer is -cos(x).