Find the derivative of `f(x)=sinx " at " x=(pi)/(2)`

Answer

To find the derivative of the function \( f(x) = \sin x \) at the point \( x = \frac{\pi}{2} \), we can use the definition of the derivative. The derivative of a function \( f \) at a point \( a \) is defined as:

\[ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \]

### Step 1: Identify the function and the point
Here, we have:
- \( f(x) = \sin x \)
- \( a = \frac{\pi}{2} \)

### Step 2: Calculate \( f(a) \)
We need to calculate \( f\left(\frac{\pi}{2}\right) \):
\[ f\left(\frac{\pi}{2}\right) = \sin\left(\frac{\pi}{2}\right) = 1 \]

### Step 3: Set up the limit for the derivative
Now we can set up the limit:
\[ f'\left(\frac{\pi}{2}\right) = \lim_{x \to \frac{\pi}{2}} \frac{\sin x - 1}{x - \frac{\pi}{2}} \]

### Step 4: Evaluate the limit
Substituting \( x = \frac{\pi}{2} \) directly into the limit gives us the indeterminate form \( \frac{0}{0} \). Therefore, we will apply L'Hôpital's Rule, which states that if we have an indeterminate form \( \frac{0}{0} \), we can differentiate the numerator and the denominator separately.

### Step 5: Differentiate the numerator and denominator
Differentiating the numerator:
\[ \frac{d}{dx}(\sin x - 1) = \cos x \]

Differentiating the denominator:
\[ \frac{d}{dx}(x - \frac{\pi}{2}) = 1 \]

### Step 6: Apply L'Hôpital's Rule
Now we can apply L'Hôpital's Rule:
\[ f'\left(\frac{\pi}{2}\right) = \lim_{x \to \frac{\pi}{2}} \frac{\cos x}{1} \]

### Step 7: Evaluate the limit
Now substituting \( x = \frac{\pi}{2} \):
\[ f'\left(\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right) = 0 \]

### Final Answer
Thus, the derivative of \( f(x) = \sin x \) at \( x = \frac{\pi}{2} \) is:
\[ f'\left(\frac{\pi}{2}\right) = 0 \]

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