Find y prime for the following equation.
yequals4 RootIndex 5 StartRoot left parenthesis x squared plus 1 right parenthesis cubed EndRoot
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Part 1
y primeequals
  


We are given the equation:

\[
y = 4 \sqrt[5]{\left( x^2 + 1 \right)^3}
\]

We need to find \( y' \), the derivative of \( y \) with respect to \( x \).

### Step 1: Rewrite the expression
First, express the fifth root and square root in terms of powers:

\[
y = 4 \left( \left( x^2 + 1 \right)^3 \right)^{1/5} = 4 \left( x^2 + 1 \right)^{3/5}
\]

### Step 2: Apply the chain rule
Now, differentiate using the chain rule. We will use the following rules:

- Power rule: \( \frac{d}{dx} \left( u^n \right) = n u^{n-1} \frac{du}{dx} \)
- Chain rule for composite functions

Let \( u = x^2 + 1 \), so that \( y = 4 u^{3/5} \).

First, differentiate \( y = 4 u^{3/5} \):

\[
y' = 4 \cdot \frac{3}{5} u^{\frac{3}{5} - 1} \cdot \frac{du}{dx}
\]

Simplifying:

\[
y' = \frac{12}{5} u^{-\frac{2}{5}} \cdot \frac{du}{dx}
\]

### Step 3: Differentiate \( u = x^2 + 1 \)
Since \( u = x^2 + 1 \), we differentiate it:

\[
\frac{du}{dx} = 2x
\]

### Step 4: Substitute back
Now substitute \( u = x^2 + 1 \) and \( \frac{du}{dx} = 2x \) into the expression for \( y' \):

\[
y' = \frac{12}{5} \left( x^2 + 1 \right)^{-\frac{2}{5}} \cdot 2x
\]

### Step 5: Simplify
Simplifying further:

\[
y' = \frac{24x}{5} \left( x^2 + 1 \right)^{-\frac{2}{5}}
\]

So the derivative \( y' \) is:

\[
y' = \frac{24x}{5} \left( x^2 + 1 \right)^{-\frac{2}{5}}
\]

Would you like more details or have any questions?

Here are five related questions to consider:
1. How does the chain rule apply to more complex functions?
2. What is the general rule for differentiating radical expressions?
3. Can you differentiate any function with a fractional exponent?
4. How do you handle products of functions in derivatives?
5. How do implicit differentiation problems differ from this one?

**Tip:** When differentiating composite functions, always identify the inner and outer functions clearly to apply the chain rule correctly.

Find y prime for the following equation: y = 4 RootIndex 5 StartRoot (x^2 + 1)^3 EndRoot.

Learn how to differentiate the equation y = 4 * (x^2 + 1)^(3/5) using the chain rule and power rule. This step-by-step solution breaks down the differentiation process, covering composite functions and radical expressions. Suitable for undergraduate calculus students.

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