Question #4b5a4

1 Answer
Jan 30, 2017

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)