Math Problem Statement

Find the derivative of the function. y = 3 √x + 6

Solution

To find the derivative of the function y=3x+6y = \frac{3}{\sqrt{x}} + 6, we need to apply the rules of differentiation.

The function can be rewritten as:

y=3x12+6y = 3x^{-\frac{1}{2}} + 6

Now, we'll differentiate term by term:

  1. The derivative of 3x123x^{-\frac{1}{2}} with respect to xx is: ddx(3x12)=3(12)x121=32x32\frac{d}{dx} \left( 3x^{-\frac{1}{2}} \right) = 3 \cdot \left(-\frac{1}{2}\right) \cdot x^{-\frac{1}{2} - 1} = -\frac{3}{2}x^{-\frac{3}{2}}

  2. The derivative of the constant 66 is: ddx(6)=0\frac{d}{dx}(6) = 0

So, the derivative of the function yy is:

dydx=32x32\frac{dy}{dx} = -\frac{3}{2}x^{-\frac{3}{2}}

This can also be expressed as:

dydx=32x3\frac{dy}{dx} = -\frac{3}{2\sqrt{x^3}}

Would you like any additional details or have any questions? Here are five related questions you might consider:

  1. What is the second derivative of y=3x+6y = \frac{3}{\sqrt{x}} + 6?
  2. How does the derivative behave as xx approaches zero?
  3. What is the slope of the function at a specific point, say x=4x = 4?
  4. How would you integrate 32x3-\frac{3}{2\sqrt{x^3}} to find the original function?
  5. Can you explore the concavity of the function y=3x+6y = \frac{3}{\sqrt{x}} + 6 using the second derivative?

Tip: Always remember that constants like 6 in differentiation disappear because the rate of change of a constant is zero.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Power rule
Constant rule

Formulas

Power rule of differentiation

Theorems

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Suitable Grade Level

Advanced High School / College