Question

Differentiate each of the following with respect to x.
1.y=x/2+2/x-5/x²

2.y=x²+5x+1/3x​

Answer 1

ANSWER:

To Differentiate:

• y = x/2 + 2/x - 5/x²
• y = x² + 5x + 1/3x

Solution:

$\bf{1) y = \dfrac{x}{2} + \dfrac{2}{x} - \dfrac{5}{x^2}} \\\\\sf{:\implies \dfrac{dy}{dx} = \dfrac{d}{dx} \left[\dfrac{x}{2} + \dfrac{2}{x} - \dfrac{5}{x^2}\right]} \\\\\sf{:\implies \dfrac{dy}{dx} = \dfrac{1}{2}\cdot \dfrac{d}{dx}[x] + 2\cdot \dfrac{d}{dx}\left[\dfrac{1}{x}\right] - 5\cdot \dfrac{d}{dx}\left[\dfrac{1}{x^2}\right]} \\\\\sf{:\implies \dfrac{dy}{dx} = \dfrac{1}{2}\cdot 1 + 2\cdot \left[\dfrac{-1}{x^2}\right] - 5\cdot \left[\dfrac{-2}{x^3}\right]} \\\\\bf{:\implies \dfrac{dy}{dx} = \dfrac{1}{2} - \dfrac{2}{x^2} + \dfrac{10}{x^3}}$

$\bf{2) y = x^2 + 5x + \dfrac{1}{3x}} \\\\\sf{:\implies \dfrac{dy}{dx} = \dfrac{d}{dx} \left[x^2 + 5x + \dfrac{1}{3x}\right]} \\\\\sf{:\implies \dfrac{dy}{dx} = \dfrac{d}{dx}[x^2] + 5\cdot \dfrac{d}{dx}[x] + \dfrac{1}{3}\cdot \dfrac{d}{dx}\left[\dfrac{1}{x}\right]} \\\\\sf{:\implies \dfrac{dy}{dx} = 2x + 5\cdot 1 + \dfrac{1}{3} \cdot \left[\dfrac{-1}{x^2}\right]} \\\\\bf{:\implies \dfrac{dy}{dx} = 2x + 5 - \dfrac{1}{3x^2}}$

Formulae Used:

• d/dx (x) = 1
• d/dx (1/x) = -1/x²
• d/dx (1/x²) = -2/x³