Question

Question #4b5a4

Answer 1

`cos(sqrtx) - (xsin(sqrtx))/(2sqrtx)`

Explanation:

I'm pretty sure you're looking for the derivative, seeing as this is posted under "Differentiating Trigonometric Functions".

Let `f(x) = xcos(sqrtx)`. This is a product of the functions

`x`, and `cos(sqrtx)`. So, using the product rule:

`(df)/dx = (x)' cos(sqrtx) + x[cos(sqrtx)]'`

`=cos(sqrtx) + x[cos(sqrtx)]'`.

Now we need to find the derivative of `cos(sqrtx)`:

The chain rule states that, if `y` is a function of `u`, and `u` is a function of `x`, then

`(dy)/(dx) = (dy)/(du) (du)/(dx)`.

Let `y = cos(sqrtx)` and `u = sqrtx`. Then `y = cosu`:

`(dy)/(dx) = -sinu * u' = -sinu * 1/(2sqrtx) = -(sin(sqrtx))/(2sqrtx)`

Therefore,

`(df)/(dx) = cos(sqrtx) - (xsin(sqrtx))/(2sqrtx)`