Question

question
find $\frac{d}{dx}(-5x^{4/3}-2x^{-5/4}-4)$
provide your answer below:
$\frac{d}{dx}(-5x^{4/3}-2x^{-5/4}-4)=$

Explanation:

Step1: Apply power - rule of differentiation

The power - rule states that $\frac{d}{dx}(x^n)=nx^{n - 1}$. For the function $y=-5x^{\frac{4}{3}}-2x^{-\frac{5}{4}}-4$, we differentiate each term separately.
For the first term $-5x^{\frac{4}{3}}$, using the power - rule, we have $\frac{d}{dx}(-5x^{\frac{4}{3}})=-5\times\frac{4}{3}x^{\frac{4}{3}-1}=-\frac{20}{3}x^{\frac{1}{3}}$.
For the second term $-2x^{-\frac{5}{4}}$, using the power - rule, we get $\frac{d}{dx}(-2x^{-\frac{5}{4}})=-2\times(-\frac{5}{4})x^{-\frac{5}{4}-1}=\frac{5}{2}x^{-\frac{9}{4}}$.
For the constant term $-4$, since $\frac{d}{dx}(c) = 0$ where $c$ is a constant, $\frac{d}{dx}(-4)=0$.

Step2: Combine the derivatives of each term

The derivative of the entire function $y=-5x^{\frac{4}{3}}-2x^{-\frac{5}{4}}-4$ is $\frac{d}{dx}(-5x^{\frac{4}{3}}-2x^{-\frac{5}{4}}-4)=-\frac{20}{3}x^{\frac{1}{3}}+\frac{5}{2}x^{-\frac{9}{4}}$.

Answer:

$-\frac{20}{3}x^{\frac{1}{3}}+\frac{5}{2}x^{-\frac{9}{4}}$