Question

calculate the derivative of y = sqrt 7x + 2

Answer 1

$y= \sqrt{7x} + 2$

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

Let's find the derivative of $\sqrt{7x}$
Let u=7x. The derivative of u, du, is 7.

$\sqrt{u}$

Take the derivative of this respect to u.

$\dfrac{1 \times du}{2 \sqrt{u} }$

du is 7.

$\dfrac{7}{2 \sqrt{u} }$

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.

$\dfrac{dy}{dx}= \dfrac{7}{2 \sqrt{u} }$

Answer 2

Answer:

$\displaystyle \frac{dy}{dx} = \frac{7}{2\sqrt{7x + 2}}$

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:                                                           $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]:                                                         $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$  

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Chain Rule]:                                                                                 $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle y = \sqrt{7x + 2}$

Step 2: Differentiate

1. Basic Power Rule:                                                                                         $\displaystyle y' = \frac{1}{2\sqrt{7x + 2}} \cdot \frac{d}{dx}[7x + 2]$
2. Basic Power Rule [Derivative Properties]:                                                   $\displaystyle y' = \frac{7}{2\sqrt{7x + 2}}$

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

Unit: Differentiation