This section covers the fundamental concepts and practice problems related to derivatives in calculus. Understanding derivatives is crucial for analyzing the behavior of functions and solving real-world problems.

Key Concepts

  • Definition of Derivative: The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point.
  • Notation: ( f'(x) ) or ( \frac{dy}{dx} )
  • Types of Derivatives:
    • Power Rule: ( \frac{d}{dx}(x^n) = nx^{n-1} )
    • Product Rule: ( \frac{d}{dx}(uv) = u'v + uv' )
    • Quotient Rule: ( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{vu' - uv'}{v^2} )

Practice Problems

Here are some practice problems to help you understand derivatives better:

  1. Find the derivative of ( f(x) = 3x^2 + 2x + 1 ).
  2. Differentiate ( g(x) = x^3 \cdot x^4 ).
  3. Calculate the derivative of ( h(x) = \frac{2x + 3}{x - 1} ).

Further Reading

For more detailed information and practice problems, you can check out our Calculus Guide.

Calculus Derivatives