https://file.aiquickdraw.com/mathbot/file/1727441358960pxov9pre.jpg

The given equation is:

\[
y = \frac{1}{(5x^3 - 3)^5}
\]

To differentiate this, we will apply the **chain rule** and the **power rule**. Let's go step-by-step through this process.

### Step 1: Identify the outer function
We have an outer function \( f(u) = u^{-5} \), where \( u = 5x^3 - 3 \).

### Step 2: Differentiate the outer function
The derivative of \( f(u) = u^{-5} \) is:

\[
f'(u) = -5u^{-6}
\]

### Step 3: Differentiate the inner function
The inner function is \( u = 5x^3 - 3 \), and its derivative is:

\[
\frac{du}{dx} = 15x^2
\]

### Step 4: Apply the chain rule
Now, using the chain rule, the derivative of \( y \) with respect to \( x \) is:

\[
\frac{dy}{dx} = -5(5x^3 - 3)^{-6} \cdot 15x^2
\]

### Step 5: Simplify the expression
Simplifying this result gives:

\[
\frac{dy}{dx} = -75x^2(5x^3 - 3)^{-6}
\]

This is the derivative of the function.

Would you like further details or have any questions?

---

Here are some related questions to explore:

1. How does the chain rule work when applied to more complex functions?
2. What are the steps for finding higher-order derivatives of this function?
3. Can this derivative be used to find the slope of a tangent at specific points?
4. How would you integrate this function instead of differentiating it?
5. How does the behavior of this function change for large values of \( x \)?

**Tip:** Always check whether simplification is possible after applying differentiation to reduce the complexity of the expression.

Differentiate the function y = 1 / (5x^3 - 3)^5

Learn how to differentiate the function y = 1 / (5x^3 - 3)^5 using the chain rule and power rule. This solution walks through each step, from identifying the inner and outer functions to applying the chain rule. Suitable for high school calculus students.

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