Find y double prime.
yequalsx Superscript six sevenths Baseline plus 2 x


The given equation is:

\[
y = x^{\frac{6}{7}} + 2x
\]

To find \( y'' \), the second derivative of \( y \) with respect to \( x \), we proceed step by step:

### Step 1: First Derivative \( y' \)
We apply the power rule \( \frac{d}{dx} \left( x^n \right) = n x^{n-1} \).

\[
y = x^{\frac{6}{7}} + 2x
\]

For \( x^{\frac{6}{7}} \), the derivative is:

\[
\frac{d}{dx} \left( x^{\frac{6}{7}} \right) = \frac{6}{7} x^{\frac{6}{7} - 1} = \frac{6}{7} x^{-\frac{1}{7}}
\]

For \( 2x \), the derivative is:

\[
\frac{d}{dx} \left( 2x \right) = 2
\]

Thus, the first derivative \( y' \) is:

\[
y' = \frac{6}{7} x^{-\frac{1}{7}} + 2
\]

### Step 2: Second Derivative \( y'' \)
Now, differentiate \( y' \) to find \( y'' \).

\[
y' = \frac{6}{7} x^{-\frac{1}{7}} + 2
\]

The derivative of \( \frac{6}{7} x^{-\frac{1}{7}} \) is:

\[
\frac{d}{dx} \left( \frac{6}{7} x^{-\frac{1}{7}} \right) = \frac{6}{7} \left( -\frac{1}{7} \right) x^{-\frac{1}{7} - 1} = -\frac{6}{49} x^{-\frac{8}{7}}
\]

The derivative of the constant \( 2 \) is:

\[
\frac{d}{dx} \left( 2 \right) = 0
\]

Thus, the second derivative \( y'' \) is:

\[
y'' = -\frac{6}{49} x^{-\frac{8}{7}}
\]

### Final Answer:
\[
y'' = -\frac{6}{49} x^{-\frac{8}{7}}
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you find the third derivative of a function?
2. How would the second derivative change if the exponent were different?
3. What does the second derivative tell us about the concavity of the function?
4. How do derivatives apply to real-world problems?
5. Can the chain rule be used in finding second derivatives?

**Tip:** Always simplify expressions before taking higher-order derivatives.

Find y double prime. y equals x to the six-sevenths plus 2x

Learn how to find the second derivative of the function y = x^(6/7) + 2x using the power rule. This step-by-step solution includes the first derivative, applying the power rule, and simplifying the expression. Suitable for students in Grades 11-12.

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