Find the derivative of the function.
yequals6StartRoot x EndRootplus5x Superscript three sevenths



To find the derivative of the function:

\[
y = 6\sqrt{x} + 5x^{\frac{3}{7}}
\]

### Step-by-Step Derivative Calculation:

1. **First term: \( 6\sqrt{x} \)**
   - Rewrite \( \sqrt{x} \) as \( x^{1/2} \).
   - Now the term becomes \( 6x^{\frac{1}{2}} \).
   - Use the power rule for derivatives: \( \frac{d}{dx}(x^n) = nx^{n-1} \).
   - Derivative of \( 6x^{\frac{1}{2}} \): 
     \[
     6 \cdot \frac{1}{2} x^{\frac{1}{2} - 1} = 3x^{-\frac{1}{2}} = \frac{3}{\sqrt{x}}
     \]

2. **Second term: \( 5x^{\frac{3}{7}} \)**
   - Use the power rule for derivatives again: \( \frac{d}{dx}(x^n) = nx^{n-1} \).
   - Derivative of \( 5x^{\frac{3}{7}} \): 
     \[
     5 \cdot \frac{3}{7} x^{\frac{3}{7} - 1} = \frac{15}{7}x^{-\frac{4}{7}}
     \]

### Final Derivative:
\[
\frac{dy}{dx} = \frac{3}{\sqrt{x}} + \frac{15}{7}x^{-\frac{4}{7}}
\]

Would you like more details or have any questions?

Here are 5 related questions to further explore this concept:
1. What is the derivative of \( \sqrt{x^2 + 1} \)?
2. How would the derivative change if the power of \( x \) in the second term was different?
3. Can you find the second derivative of this function?
4. What is the integral of \( 6\sqrt{x} + 5x^{3/7} \)?
5. How do you apply the chain rule when differentiating composite functions?

**Tip:** When working with fractional exponents, it helps to rewrite radical expressions as powers to apply the power rule directly.

Find the derivative of the function: y = 6√x + 5x^(3/7)

Learn how to find the derivative of the function y = 6√x + 5x^(3/7) using the power rule. This detailed solution explains each step, from rewriting radicals as exponents to applying the power rule. Suitable for high school students studying calculus in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation