Question

Differentiate Vx(x + 2) with respect to x.

Answer 1

`y'= (3x+2)/(2√(x) )`

General Formulas and Concepts:

Calculus

The derivative of a constant is equal to 0

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Product Rule:
`(d)/(dx) [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)`

Chain Rule:
`(d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

Explanation:

Step 1: Define

`y=√(x) (x+2)`

Step 2: Rewrite

`y=x^{(1)/(2) } (x+2)`

Step 3: Differentiate

1. Product Rule [Basic Power/Chain Rule]:
   `y'= (1)/(2) x^{(1)/(2) -1}(x+2)+x^{(1)/(2) }(1)`
2. Simplify:
   `y'= (1)/(2) x^{(-1)/(2)}(x+2)+x^{(1)/(2)}`
3. Rewrite:
   `y'= (x+2)/(2√(x) ) + √(x)`
4. Add:
   `y'= (3x+2)/(2√(x) )`