Question 63931
I am assuming that the fractions after the variables are exponents, let me know if I'm not interpretting you correctly:
calculate the first derivatives f(x) for each of the following:
show all work

A)f(x)= 2/3x3/4 +  3/4x2/3+ 5/6x2/3
{{{f(x)=(2/3)x^(3/4)+(3/4)x^(2/3)+(5/6)x^(2/3)}}}
{{{f(x)=(2/3)x^(3/4)+(3/4+5/6)x^(2/3)}}}
{{{f(x)=(2/3)x^(3/4)+(9/12+10/12)x^(2/3)}}}
{{{f(x)=(2/3)x^(3/4)+(19/12)x^(2/3)}}}
f'(x)={{{(3/4)(2/3)x^(3/4-1)+(2/3)(19/12)x^(2/3-1)}}}
f'(x)={{{(1/2)x^(-1/4)+(19/18)x^(-1/3)}}}
f'(x)={{{1/2x^(1/4)+19/18x^(1/3)}}}
:
B) f(x)= 2/5x3/2 + 5/6x3/4 + 5/6x3/2
{{{f(x)=(2/5)x^(3/2)+(5/6)x^(3/4)+(5/6)x^(3/2)}}}
{{{f(x)=(2/5+5/6)x^(3/2)+(5/6)x^(3/4)}}}
{{{f(x)=(12/30+25/30)x^(3/2)+(5/6)x^(3/4)}}}
{{{f(x)=(27/30)x^(3/2)+(5/6)x^(3/4)}}}
f'(x)={{{(3/2)(37/30)x^(3/2-1)+(3/4)(5/6)x^(3/4-1)}}}
f'(x)={{{(37/20)x^(1/2)+(5/8)x^(-1/4)}}}
f'(x)={{{37*sqrt(x)/20+5/8^(1/4)}}}
:

C) f(x)= 3/4x2/3 +5/6x3/2 + 3/4x3/2
{{{f(x)=(3/4)x^(2/3)+(5/6)x^(3/2)+(3/4)x^(3/2)}}}
{{{f(x)=(3/4)x^(2/3)+(5/6+3/4)x^(3/2)}}}
{{{f(x)=(3/4)x^(2/3)+(19/12)x^(3/2)}}}
f'(x)={{{(2/3)(3/4)x^(2/3-1)+(3/2)(19/12)x^(3/2-1)}}}
f'(x)={{{(1/2)x^(-1/3)+(19/8)x^(1/2)}}}
f'(x)={{{1/2x^(1/3)+19sqrt(x)/8}}}
Happy Calculating!!!