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To find the gradient of the curve \( y = \frac{1}{1 + \cos x} \) at the point \( \left( \frac{\pi}{2}, 1 \right) \), we need to calculate the derivative \( \frac{dy}{dx} \) and evaluate it at \( x = \frac{\pi}{2} \).

### Steps:

1. **Differentiate** \( y = \frac{1}{1 + \cos x} \) using the quotient rule.

The quotient rule states that for \( y = \frac{u(x)}{v(x)} \), the derivative is:

\[
\frac{dy}{dx} = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{[v(x)]^2}
\]

In this case:
- \( u(x) = 1 \), so \( u'(x) = 0 \)
- \( v(x) = 1 + \cos x \), so \( v'(x) = -\sin x \)

Thus, applying the quotient rule:

\[
\frac{dy}{dx} = \frac{(1 + \cos x) \cdot 0 - 1 \cdot (-\sin x)}{(1 + \cos x)^2}
\]

Simplifying:

\[
\frac{dy}{dx} = \frac{\sin x}{(1 + \cos x)^2}
\]

2. **Evaluate the derivative at \( x = \frac{\pi}{2} \):**

At \( x = \frac{\pi}{2} \), we know:
- \( \sin \left( \frac{\pi}{2} \right) = 1 \)
- \( \cos \left( \frac{\pi}{2} \right) = 0 \)

Substituting these values into the derivative:

\[
\frac{dy}{dx} = \frac{1}{(1 + 0)^2} = \frac{1}{1} = 1
\]

Thus, the gradient of the curve at the point \( \left( \frac{\pi}{2}, 1 \right) \) is **1**.

Would you like more details, or do you have any questions about this solution?

### Related Questions:
1. How would the gradient change if the function were \( y = \frac{1}{1 + \sin x} \)?
2. Can you find the second derivative \( \frac{d^2y}{dx^2} \) of the given function?
3. What is the gradient of the curve \( y = \tan(x) \) at \( x = \frac{\pi}{4} \)?
4. How can the chain rule be applied to differentiate more complex trigonometric functions?
5. How does the behavior of the curve \( y = \frac{1}{1 + \cos x} \) change as \( x \) approaches \( 0 \)?

### Tip:
When applying the quotient rule, always simplify both the numerator and denominator before substituting values to reduce calculation errors.

Find the gradient of the curve y = 1 / (1 + cos(x)) at the point (π/2, 1). (Give your answer as an exact whole number or as a fraction in its lowest terms in the form p/q).

Learn how to find the gradient of the curve y = 1 / (1 + cos(x)) at the point (π/2, 1) using differentiation techniques like the quotient rule. This problem involves calculus and trigonometric functions, suitable for students in Grades 11-12.

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