What is the derivative of the function f(x) = (1/2)x⁶ + 3x^(4/3) + 4x with respect to x?

To find the derivative of the function ( f(x) = \frac{1}{2}x^6 + 3x^{4/3} + 4x ), we apply the power rule of differentiation, which states that the derivative of ( x^n ) is ( nx^{n-1} ).

1. For the term ( \frac{1}{2}x^6 ), applying the power rule gives:

[

\frac{d}{dx}\left(\frac{1}{2}x^6\right) = 6 \cdot \frac{1}{2}x^{6-1} = 3x^5

]

2. For the term ( 3x^{4/3} ), again using the power rule:

[

\frac{d}{dx}(3x^{4/3}) = 3 \cdot \frac{4}{3}x^{4/3 - 1} = 4x^{1/3}

]

3. For the term ( 4x ):

[

\frac{d}{dx}(4x) = 4

\