Differentiate.
yequals=StartFraction x Over 5 EndFraction
x
5minus−StartFraction 5 Over x EndFraction
5
x

Let's differentiate the given expression step by step. The function you provided is:

\[
y = \frac{x}{5} \cdot x^5 - \frac{5}{x} \cdot x^5
\]

### Step 1: Simplify the expression
We'll first simplify the given expression.

\[
y = \frac{x}{5} \cdot x^5 = \frac{x^6}{5}
\]

\[
y = \frac{5}{x} \cdot x^5 = 5 \cdot x^4
\]

So, the simplified function is:

\[
y = \frac{x^6}{5} - 5x^4
\]

### Step 2: Differentiate
We now differentiate each term.

- For the first term \( \frac{x^6}{5} \), we apply the power rule:
  \[
  \frac{d}{dx} \left( \frac{x^6}{5} \right) = \frac{6x^5}{5}
  \]

- For the second term \( -5x^4 \), we also apply the power rule:
  \[
  \frac{d}{dx} \left( -5x^4 \right) = -20x^3
  \]

### Final result:
The derivative of \( y \) is:

\[
y' = \frac{6x^5}{5} - 20x^3
\]

### Do you want more details on any step or have questions?

---

Here are 5 related questions for you:

1. How do you differentiate other power functions like \( x^n \)?
2. What is the power rule for differentiation?
3. Can you differentiate trigonometric functions like \( \sin(x) \) or \( \cos(x) \)?
4. What are the basic rules of differentiation (e.g., product rule, chain rule)?
5. How do you integrate functions involving powers of \( x \)?

**Tip**: Always simplify expressions before differentiating to make the process easier and avoid errors!

Differentiate: y = (x/5) * x^5 - (5/x) * x^5

Learn how to differentiate the function y = (x/5) * x^5 - (5/x) * x^5 with a step-by-step solution. This tutorial simplifies the function and applies the power rule of differentiation. Suitable for students in Grades 11-12 or introductory college-level calculus.

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