1. **State the problem:** We want to determine if a function is differentiable at $x=\frac{\pi}{2}$ using the definition of the derivative. 2. **Recall the definition of differentiability:** A function $f(x)$ is differentiable at $x=a$ if the limit $$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$ exists and is finite. 3. **Apply the definition at $x=\frac{\pi}{2}$:** We need to evaluate $$\lim_{h \to 0} \frac{f\left(\frac{\pi}{2} + h\right) - f\left(\frac{\pi}{2}\right)}{h}$$ 4. **Important rules:** - The function must be defined at $x=\frac{\pi}{2}$. - The limit must exist and be the same from both sides (left and right). 5. **Intermediate work:** - Substitute $x=\frac{\pi}{2}$ into the function. - Calculate $f\left(\frac{\pi}{2} + h\right)$. - Simplify the difference quotient. - Check if the limit as $h \to 0$ exists. 6. **Conclusion:** - If the limit exists and is finite, $f$ is differentiable at $x=\frac{\pi}{2}$. - Otherwise, it is not differentiable there. Since the function is not specified, this is the general method to check differentiability at $x=\frac{\pi}{2}$ using the definition.