Question

1. find $\frac{dy}{dx}$ for a) $y=\frac{5}{4}x^{4}-\frac{5}{x^{2}}+sqrt3{x^{2}}+pi$

Explanation:

Step1: Rewrite terms

Rewrite the function $y=\frac{5}{4}x^{4}-\frac{5}{x^{2}}+\sqrt[3]{x^{2}}+\pi$ as $y = \frac{5}{4}x^{4}-5x^{- 2}+x^{\frac{2}{3}}+\pi$.

Step2: Apply power - rule

The power - rule for differentiation is $\frac{d}{dx}(x^{n})=nx^{n - 1}$.
For the first term $\frac{5}{4}x^{4}$, $\frac{d}{dx}(\frac{5}{4}x^{4})=\frac{5}{4}\times4x^{4 - 1}=5x^{3}$.
For the second term $-5x^{-2}$, $\frac{d}{dx}(-5x^{-2})=-5\times(-2)x^{-2 - 1}=10x^{-3}$.
For the third term $x^{\frac{2}{3}}$, $\frac{d}{dx}(x^{\frac{2}{3}})=\frac{2}{3}x^{\frac{2}{3}-1}=\frac{2}{3}x^{-\frac{1}{3}}$.
For the fourth term $\pi$ (a constant), $\frac{d}{dx}(\pi)=0$.

Step3: Combine the derivatives

$\frac{dy}{dx}=5x^{3}+10x^{-3}+\frac{2}{3}x^{-\frac{1}{3}}$.

Answer:

$\frac{dy}{dx}=5x^{3}+\frac{10}{x^{3}}+\frac{2}{3\sqrt[3]{x}}$