Work Check: Differentiate f[x] = (1+x)^(3/2) (2+x)^(2/3)

[confused_07]

Can you please check my work.....
Differentiate the function f[x] = (1+x)^(3/2) (2+x)^(2/3)

I first found the derivatives of each using the power rule:

h'[x]= (1+x)^(3/2) = (3/2)(1+x)^(1/2)
g'[x]= (2+x)^(2/3) = (2/3)(2+x)^(-1/3)

Then I used the product rule to get:

f'[x]= [(1+x)^(3/2) * (2/3)(2+x)^(-1/3)] + [(3/2)(1+x)^(1/2) * (2+x)^(2/3)]

Is this correct or did I hose this all up?

 

[galactus]

You're correct. Now try to simplify it into something nice.

Maybe:

\(\displaystyle \L\\\frac{sqrt{x+1}(13x+22)}{6(x+2)^{\frac{1}{3}}}\)