Verified Answer :

Preetham123

Answer:

To find the derivative of a square root, you will just need to consider the expression under the square root as a whole term, or in calculus, as a variable called 'u.'

In this case, our expression is: √(5x-6)

Let u = 5x - 6

du = 5

d/du (√u) = du 0.5 u^(-1/2)

Substitute back for u:

d/dx (√u) = 5 0.5 (5x - 6)^(-1/2)

d/dx (√u) = 2.5/√(5x - 6)

Hope this helped!

Space

Answer:

\(\displaystyle \frac{d}{dx}[\sqrt{5x - 6}] = \frac{5}{2\sqrt{5x - 6}}\)

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

\(\displaystyle y = \sqrt{5x - 6}\)

Step 2: Differentiate

  1. [Function] Rewrite:                                                                                        \(\displaystyle y = (5x - 6)^\Big{\frac{1}{2}}\)
  2. [Function] Basic Power Rule [Derivative Rule - Chain Rule]:                     \(\displaystyle y' = \frac{1}{2}(5x - 6)^\Big{-\frac{1}{2}} \frac{d}{dx}[5x - 6]\)
  3. Rewrite:                                                                                                         \(\displaystyle y' = \frac{1}{2(5x - 6)^\Big{\frac{1}{2}}}\frac{d}{dx}[5x - 6]\)
  4. Rewrite [Derivative Rule - Addition/Subtraction]:                                       \(\displaystyle y' = \frac{1}{2(5x - 6)^\Big{\frac{1}{2}}} \bigg( \frac{d}{dx}[5x] - \frac{d}{dx}[6] \bigg)\)
  5. Rewrite [Derivative Rule - Multiplied Constant]:                                         \(\displaystyle y' = \frac{1}{2(5x - 6)^\Big{\frac{1}{2}}} \bigg( 5 \frac{d}{dx}[x] - \frac{d}{dx}[6] \bigg)\)
  6. Basic Power Rule:                                                                                         \(\displaystyle y' = \frac{5}{2(5x - 6)^\Big{\frac{1}{2}}}\)
  7. Rewrite:                                                                                                         \(\displaystyle y' = \frac{5}{2\sqrt{5x - 6}}\)

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

Unit: Differentiation

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