Question

question 14
find: $\frac{d}{dx}(\frac{4}{x^{4}} - 5sqrt3{x})$
no correct answer choice is given.
$\frac{1}{4x^{3}}-\frac{5}{3}x^{-2/3}$
$\frac{1}{x^{3}}+\frac{5}{3}x^{-4/3}$
$-16x^{-5}-\frac{5}{3}x^{-2/3}$
$\frac{1}{4}x^{-5}-15x^{2/3}$

Explanation:

Step1: Rewrite the function

Rewrite $\frac{4}{x^{4}}-5\sqrt[3]{x}$ as $4x^{-4}-5x^{\frac{1}{3}}$.

Step2: Apply the power - rule for differentiation

The power - rule states that $\frac{d}{dx}(ax^{n})=nax^{n - 1}$. For $y = 4x^{-4}$, $\frac{d}{dx}(4x^{-4})=4\times(-4)x^{-4 - 1}=-16x^{-5}$. For $y=-5x^{\frac{1}{3}}$, $\frac{d}{dx}(-5x^{\frac{1}{3}})=-5\times\frac{1}{3}x^{\frac{1}{3}-1}=-\frac{5}{3}x^{-\frac{2}{3}}$.

Step3: Combine the derivatives

The derivative of the sum/difference of two functions is the sum/difference of their derivatives. So, $\frac{d}{dx}(4x^{-4}-5x^{\frac{1}{3}})=-16x^{-5}-\frac{5}{3}x^{-\frac{2}{3}}$.

Answer:

$-16x^{-5}-\frac{5}{3}x^{-\frac{2}{3}}$ (corresponding to the fourth option in the multiple - choice list)