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)#