Common Derivatives Formulas

Understanding common derivative formulas is key in calculus and statistics methods. These rules simplify differentiation, helping to analyze functions' behavior, slopes, and rates of change. Mastering them lays a strong foundation for tackling more complex mathematical concepts.

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 the sum or difference of two functions.
   • Each function can be differentiated independently.
   • Facilitates the analysis of combined functions.

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 functions that are products of variables.

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 rational functions and their behavior.

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 understanding nested functions and their rates of change.

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

   • The derivative of the natural exponential function is itself.
   • Highlights the unique property of the exponential function.
   • Fundamental in growth and decay models in calculus.

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

   • Derivative is the reciprocal of x, valid for x > 0.
   • Important for solving problems involving logarithmic growth.
   • Connects exponential and logarithmic functions.

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 shows the relationship with secant function.

10. Inverse Trigonometric Functions:

    • d/dx(arcsin(x)) = 1 / sqrt(1 - x^2)
      • Derivative is defined for -1 < x < 1.
    • d/dx(arccos(x)) = -1 / sqrt(1 - x^2)
      • Negative derivative reflects the decreasing nature of arccos.
    • d/dx(arctan(x)) = 1 / (1 + x^2)
      • Derivative indicates the slope of the arctangent function.