Question

Calculate the derivative of y = sqrt 7x + 2

Answer 1

`y= √(7x) + 2`

Let's solve this by taking the derivative of each term in the equation.

Let's find the derivative of
`√(7x)`
Let u=7x. The derivative of u, du, is 7.


`√(u)`

Take the derivative of this respect to u.


`(1 * du)/(2 √(u) )`

du is 7.


`(7)/(2 √(u) )`

That's the derivative of the first term.
Finding the derivative of the second term is easy. The derivative of a constant is 0. Thus, the derivative of 2 is 0.

Put the two derivatives of each of the terms back into the equation to get the derivative of the whole function.


`(dy)/(dx)= (7)/(2 √(u) )`

Answer 2

`\displaystyle (dy)/(dx) = (7)/(2√(7x + 2))`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

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

Explanation:

Step 1: Define

Identify

`\displaystyle y = √(7x + 2)`

Step 2: Differentiate

1. Basic Power Rule:
   `\displaystyle y' = (1)/(2√(7x + 2)) \cdot (d)/(dx)[7x + 2]`
2. Basic Power Rule [Derivative Properties]:
   `\displaystyle y' = (7)/(2√(7x + 2))`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation