what is the derivative of $f(x)=5x^{\frac{1}{3}} + 4x^{2}-6sqrt{x}$?\n$\frac{5}{3}x^{\frac{2}{3}}+8x…

Explanation:

Step1: Recall power - rule for derivatives

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$.

Step2: Differentiate each term of $f(x)$

For the first term $y_1 = 5x^{\frac{1}{3}}$, using the power - rule, $y_1^\prime=5\times\frac{1}{3}x^{\frac{1}{3}-1}=\frac{5}{3}x^{-\frac{2}{3}}$. For the second term $y_2 = 4x^2$, using the power - rule, $y_2^\prime=4\times2x^{2 - 1}=8x$. For the third term $y_3=-6\sqrt{x}=-6x^{\frac{1}{2}}$, using the power - rule, $y_3^\prime=-6\times\frac{1}{2}x^{\frac{1}{2}-1}=-3x^{-\frac{1}{2}}$.

Step3: Combine the derivatives of each term

$f^\prime(x)=y_1^\prime + y_2^\prime+y_3^\prime=\frac{5}{3}x^{-\frac{2}{3}}+8x - 3x^{-\frac{1}{2}}$.

Answer:

$\frac{5}{3}x^{-\frac{2}{3}}+8x - 3x^{-\frac{1}{2}}$ (corresponds to the fourth option $\frac{5}{3}x^{\frac{2}{3}}+8x - 3x^{-\frac{1}{2}}$ after correcting the exponent sign in the first term's exponent in the option from positive to negative as our derived first - term exponent is negative)