To derive the function \( f(t) = \sqrt{\sin(x) + 1} \), we need to find its derivative with respect to \( x \). 

Given:
\[
f(x) = \sqrt{\sin(x) + 1}
\]

We can rewrite the function using the exponent form:
\[
f(x) = (\sin(x) + 1)^{1/2}
\]

Now, we apply the chain rule for differentiation. The chain rule states that if \( y = (g(x))^n \), then:
\[
\frac{dy}{dx} = n \cdot (g(x))^{n-1} \cdot g'(x)
\]

### Step-by-step Differentiation:

1. Identify \( g(x) = \sin(x) + 1 \) and \( n = \frac{1}{2} \).
2. Compute the derivative of \( g(x) \): 
   \[
   g'(x) = \cos(x)
   \]

3. Apply the chain rule:
\[
f'(x) = \frac{1}{2} (\sin(x) + 1)^{-1/2} \cdot \cos(x)
\]

4. Simplify the expression:
\[
f'(x) = \frac{\cos(x)}{2 \sqrt{\sin(x) + 1}}
\]

### Conclusion:
The derivative of \( f(x) = \sqrt{\sin(x) + 1} \) with respect to \( x \) is:
\[
f'(x) = \frac{\cos(x)}{2 \sqrt{\sin(x) + 1}}
\]

Would you like more details or have any other questions?

### Related Questions:
1. How do you apply the chain rule in differentiation for more complex functions?
2. What is the derivative of \( \sqrt{\sin(x)} \) without the additional constant?
3. How do you differentiate functions involving square roots and trigonometric functions?
4. Can you explain the chain rule in the context of functions with higher powers?
5. How does the derivative of trigonometric functions affect their graphs?

#### Tip:
When dealing with square roots in differentiation, it's often helpful to rewrite them as exponentials to apply the power rule more easily.

Learn how to derive the function f(x) = sqrt(sin(x) + 1) using the chain rule. This detailed guide explains the process of differentiating functions involving square roots and trigonometric functions, suitable for Grades 11-12 students.

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