Question

How do you differentiate `(x+x^(1/2))(cosx+x^(1/3))`?

Answer 1

`f(x)' = (x + sqrt(x))(1/(3x^(2/3)) - sin x) + (cos x + root(3)(x))(1 + 1/(2sqrt(x)))`

Explanation:

Use the Product rule and the Power rule in combination:

Product Rule: `(u*v)' = u*v' + v*u'`
Power Rule: `(u^n)' = n*u^(n-1)*u'`

Let `u = x + x^(1/2)` so `u' = 1 + 1/2x^(-1/2)`
Let `v = cos x + x^(1/3)` so `v' = -sin x + 1/3x^(-2/3)`

`f(x)' = (x + x^(1/2))( -sin x + 1/3x^(-2/3)) + (cos x + x^(1/3))(1 + 1/2x^(-1/2))`

So:

`f(x)' = (x + sqrt(x))(1/(3x^(2/3)) - sin x) + (cos x + root(3)(x))(1 + 1/(2sqrt(x)))`