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The document you've uploaded contains calculus exercises related to derivatives and differentiability. Here’s a brief overview of the problems:

1. **Derivative using the definition**: Find the first derivative of \( f(x) = 2x^2 + x + 1 \) using the limit definition.
2. **Differentiability at a point**: Check if the given piecewise function is differentiable at \( x = 3 \), and find \( f'(3) \) if it exists.
3. **Differentiability at another point**: Similar to #2, but check differentiability at \( x = 1 \) for a different piecewise function.
4. **Finding the first derivative** for several functions, including polynomials, trigonometric, and rational functions.
5. **Second derivatives** for two functions: a trigonometric function and a polynomial.
6. **Equation of tangent and normal lines**: Find the equations of both the tangent and normal lines to \( y = (2x + 1)^3 \) at the point \( (0, 1) \).

Would you like a detailed solution for any specific question?

Here are 5 related questions you might find useful:
1. How do you derive a function using the limit definition?
2. What are the criteria for checking differentiability at a point?
3. How do you handle trigonometric functions in derivative calculations?
4. What are the steps to find the second derivative of a function?
5. How do you derive the equation of a normal line to a curve?

**Tip**: When checking differentiability at a point, always remember to evaluate both the left-hand and right-hand limits of the derivative.

The document you've uploaded contains calculus exercises related to derivatives and differentiability. The problems include calculating derivatives using the limit definition, checking differentiability at certain points, and finding tangent and normal lines equations.

This page provides detailed solutions to calculus exercises covering derivatives using the limit definition, checking differentiability at given points, and finding equations for tangent and normal lines. Key topics include first and second derivatives, and piecewise functions.

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