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).