Math Problem Statement

Differentiate the function g(u) = √(3u) + √(2u).

Solution

We are given the function:

g(u)=3u+2ug(u) = \sqrt{3u} + \sqrt{2u}

To find g(u)g'(u), we need to differentiate this expression with respect to uu.

Step-by-step solution:

  1. Rewrite the square roots as exponents: g(u)=(3u)1/2+(2u)1/2g(u) = (3u)^{1/2} + (2u)^{1/2}

  2. Apply the chain rule to differentiate each term:

    • For (3u)1/2(3u)^{1/2}, use the chain rule: ddu((3u)1/2)=12(3u)1/23=323u\frac{d}{du} \left( (3u)^{1/2} \right) = \frac{1}{2} (3u)^{-1/2} \cdot 3 = \frac{3}{2\sqrt{3u}}
    • For (2u)1/2(2u)^{1/2}, similarly: ddu((2u)1/2)=12(2u)1/22=12u\frac{d}{du} \left( (2u)^{1/2} \right) = \frac{1}{2} (2u)^{-1/2} \cdot 2 = \frac{1}{\sqrt{2u}}
  3. Combine the results: g(u)=323u+12ug'(u) = \frac{3}{2\sqrt{3u}} + \frac{1}{\sqrt{2u}}

Thus, the derivative of g(u)g(u) is:

g(u)=323u+12ug'(u) = \frac{3}{2\sqrt{3u}} + \frac{1}{\sqrt{2u}}

Let me know if you would like further clarification or have any questions!

Here are some related questions to expand your understanding:

  1. How do you apply the chain rule to more complicated functions?
  2. What happens if you differentiate a product of functions involving square roots?
  3. How would you integrate the function g(u)g(u)?
  4. What is the difference between implicit and explicit differentiation?
  5. How can the power rule be used to differentiate terms with fractional exponents?

Tip: Always check if functions can be simplified before differentiation—it often makes the process easier!

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Math Problem Analysis

Mathematical Concepts

Differentiation
Chain Rule
Square Roots

Formulas

g(u) = √(3u) + √(2u)
Derivative of a square root: d/dx (√x) = 1/(2√x)
Chain Rule

Theorems

Chain Rule for Differentiation
Power Rule for Differentiation

Suitable Grade Level

Grades 11-12