Common Derivative Formulas

Understanding common derivative formulas is key in Differential Calculus. These rules simplify the process of finding derivatives, helping to analyze functions, slopes, and rates of change. Mastering them lays a strong foundation for tackling more complex calculus problems.

1. Power Rule: d/dx(x^n) = nx^(n-1)

   • Applies to any real number exponent n.
   • Simplifies the differentiation of polynomial functions.
   • Essential for finding slopes of curves defined by power functions.

2. Constant Rule: d/dx(c) = 0

   • States that the derivative of a constant is zero.
   • Indicates that constants do not change, hence no slope.
   • Useful in simplifying expressions involving constants.

3. Sum/Difference Rule: d/dx(f(x) ± g(x)) = f'(x) ± g'(x)

   • Allows differentiation of sums and differences of functions.
   • Each function can be differentiated independently.
   • Facilitates the handling of complex expressions.

4. Product Rule: d/dx(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)

   • Used when differentiating the product of two functions.
   • Requires both functions to be differentiated and combined.
   • Important for applications involving products in calculus.

5. Quotient Rule: d/dx(f(x)/g(x)) = (f'(x)g(x) - f(x)g'(x)) / [g(x)]^2

   • Applies to the division of two functions.
   • Involves differentiating both the numerator and denominator.
   • Essential for finding derivatives of rational functions.

6. Chain Rule: d/dx(f(g(x))) = f'(g(x)) * g'(x)

   • Used for composite functions where one function is inside another.
   • Requires differentiation of the outer function and inner function.
   • Critical for handling nested functions in calculus.

7. Exponential Function: d/dx(e^x) = e^x

   • The derivative of the natural exponential function is itself.
   • Simplifies calculations involving exponential growth or decay.
   • Fundamental in many areas of mathematics and science.

8. Natural Logarithm: d/dx(ln(x)) = 1/x

   • Derivative applies only for x > 0.
   • Inversely related to the exponential function.
   • Useful in solving equations involving logarithmic expressions.

9. Trigonometric Functions:

   • d/dx(sin(x)) = cos(x)
     • Derivative represents the rate of change of sine.
   • d/dx(cos(x)) = -sin(x)
     • Indicates the negative rate of change of cosine.
   • d/dx(tan(x)) = sec^2(x)
     • Derivative relates to the secant function, important in trigonometry.

10. Inverse Trigonometric Functions:

    • d/dx(arcsin(x)) = 1 / √(1 - x^2)
      • Applies for -1 < x < 1.
    • d/dx(arccos(x)) = -1 / √(1 - x^2)
      • Also applies for -1 < x < 1.
    • d/dx(arctan(x)) = 1 / (1 + x^2)
      • Valid for all real x, useful in calculus and geometry.