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The given equation is:

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

We are asked to find the gradient (or slope) of the curve at \(x = 1\). This means we need to compute the derivative of \(y\) with respect to \(x\) and evaluate it at \(x = 1\).

### Steps:
1. Differentiate \(y = \frac{1}{(1 - 3x)^5}\) with respect to \(x\), using the chain rule.
2. Plug in \(x = 1\) into the derivative to find the gradient at that point.

I'll compute this now.The derivative of \( y = \frac{1}{(1 - 3x)^5} \) with respect to \(x\) is:

\[
\frac{dy}{dx} = \frac{15}{(1 - 3x)^6}
\]

At \(x = 1\), the gradient of the curve is:

\[
\frac{dy}{dx} \bigg|_{x=1} = \frac{15}{64}
\]

Thus, the gradient of the curve at \(x = 1\) is \( \frac{15}{64} \).

Would you like more details on any step? 

Here are five related questions:
1. How do you apply the chain rule in this differentiation?
2. Why is it important to simplify the derivative before evaluation?
3. How do you interpret the gradient of a curve geometrically?
4. What is the significance of the power rule in this context?
5. How does the steepness of the curve change as \(x\) increases?

**Tip:** Always check if a function has any undefined points before evaluating at specific values to avoid errors in calculus.

Find the gradient of the curve y = 1 / (1 - 3x)^5 at x = 1.

Learn how to find the gradient of the curve y = 1 / (1 - 3x)^5 at x = 1 using differentiation and the chain rule. This step-by-step solution covers key calculus concepts, suitable for students in Grades 11-12.

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