Derivative of a sum with fractional exponents, evaluated at a point

Apply the sum/difference and constant multiple rules to differentiate term by term: $f'(x) = 3\cdot \frac{2}{3}x^{2/3-1} - 7\cdot\frac{1}{2}x^{1/2-1} + 0$. This simplifies to $f'(x) = 2x^{-1/3} - \frac{7}{2}x^{-1/2}$.

Now substitute $x=4$: $f'(4) = 2(4)^{-1/3} - \frac{7}{2}(4)^{-1/2}$. Since $(4)^{-1/2} = \frac{1}{\sqrt{4}} = \frac{1}{2}$, we have $f'(4) = 2(4)^{-1/3} - \frac{7}{2}\cdot\frac{1}{2} = 2(4)^{-1/3} - \frac{7}{4}$. The first term is $2\cdot \frac{1}{\sqrt[3]{4}} = \frac{2}{\sqrt[3]{4}} \approx 1.260$, and the second term is $\frac{7}{4} = 1.75$, so $f'(4) \approx 1.260 - 1.75 \approx -0.49$.