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?

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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!