Using the definition of derivative find the derivative of `sqrt(sin x)`

Explanation

To find the derivative of \( f(x) = \sqrt{\sin x} \) using the definition of the derivative, we will follow these steps:

### Step 1: Write the definition of the derivative
The derivative of a function \( f(x) \) is defined as:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

### Step 2: Substitute \( f(x) \) into the definition
Here, we have:
\[ f(x) = \sqrt{\sin x} \]
Thus, we need to find \( f(x+h) \):
\[ f(x+h) = \sqrt{\sin(x+h)} \]
Now, substituting into the definition:
\[ f'(x) = \lim_{h \to 0} \frac{\sqrt{\sin(x+h)} - \sqrt{\sin x}}{h} \]

### Step 3: Rationalize the numerator
To simplify the expression, we will multiply the numerator and denominator by the conjugate of the numerator:
\[ f'(x) = \lim_{h \to 0} \frac{\left(\sqrt{\sin(x+h)} - \sqrt{\sin x}\right) \left(\sqrt{\sin(x+h)} + \sqrt{\sin x}\right)}{h \left(\sqrt{\sin(x+h)} + \sqrt{\sin x}\right)} \]
This gives us:
\[ f'(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h \left(\sqrt{\sin(x+h)} + \sqrt{\sin x}\right)} \]

### Step 4: Use the sine difference identity
We know that:
\[ \sin(x+h) - \sin x = 2 \cos\left(\frac{(x+h)+x}{2}\right) \sin\left(\frac{h}{2}\right) \]
Thus, we can rewrite the limit:
\[ f'(x) = \lim_{h \to 0} \frac{2 \cos\left(x + \frac{h}{2}\right) \sin\left(\frac{h}{2}\right)}{h \left(\sqrt{\sin(x+h)} + \sqrt{\sin x}\right)} \]

### Step 5: Simplify the limit
We can express \( \sin\left(\frac{h}{2}\right) \) in terms of \( h \):
\[ \sin\left(\frac{h}{2}\right) \approx \frac{h}{2} \text{ as } h \to 0 \]
Thus:
\[ f'(x) = \lim_{h \to 0} \frac{2 \cos\left(x + \frac{h}{2}\right) \cdot \frac{h}{2}}{h \left(\sqrt{\sin(x+h)} + \sqrt{\sin x}\right)} \]
This simplifies to:
\[ f'(x) = \lim_{h \to 0} \frac{\cos\left(x + \frac{h}{2}\right)}{\sqrt{\sin(x+h)} + \sqrt{\sin x}} \]

### Step 6: Evaluate the limit
As \( h \to 0 \):
\[ \cos\left(x + \frac{h}{2}\right) \to \cos x \]
And:
\[ \sqrt{\sin(x+h)} + \sqrt{\sin x} \to 2\sqrt{\sin x} \]
Thus:
\[ f'(x) = \frac{\cos x}{2\sqrt{\sin x}} \]

### Final Answer
The derivative of \( \sqrt{\sin x} \) is:
\[ f'(x) = \frac{\cos x}{2\sqrt{\sin x}} \]