Fundamental Calculus Formulas

Fundamental calculus formulas are essential tools in physical sciences, helping us understand how quantities change. These rules simplify differentiation and integration, making it easier to analyze functions and their behaviors in various scientific contexts.

1. Derivative of a constant: d/dx(c) = 0

   • The derivative of any constant value is always zero.
   • This indicates that constants do not change with respect to the variable x.
   • It serves as a foundational concept in calculus, simplifying the differentiation process.

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

   • This rule applies to any real number exponent n.
   • It allows for quick differentiation of polynomial functions.
   • The exponent is brought down as a coefficient, and the exponent is decreased by one.

3. Sum rule: d/dx(f(x) + g(x)) = f'(x) + g'(x)

   • The derivative of a sum of functions is the sum of their derivatives.
   • This rule simplifies the differentiation of combined functions.
   • It emphasizes the linearity of differentiation.

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.
   • Each function is differentiated while the other remains unchanged.
   • This rule is essential for handling more complex functions.

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

   • Applies when differentiating the quotient of two functions.
   • It involves differentiating both the numerator and denominator.
   • The result is a more complex expression that requires careful handling.

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

   • This rule is crucial for differentiating composite functions.
   • It involves taking the derivative of the outer function and multiplying it by the derivative of the inner function.
   • It allows for the differentiation of nested functions.

7. Derivative of exponential function: d/dx(e^x) = e^x

   • The derivative of the natural exponential function is itself.
   • This property is unique to the base e and is fundamental in calculus.
   • It simplifies calculations involving exponential growth or decay.

8. Derivative of logarithmic function: d/dx(ln(x)) = 1/x

   • The derivative of the natural logarithm function is the reciprocal of x.
   • This rule is essential for solving problems involving logarithmic growth.
   • It highlights the relationship between logarithmic and exponential functions.

9. Derivative of trigonometric functions: d/dx(sin(x)) = cos(x), d/dx(cos(x)) = -sin(x)

   • The derivatives of sine and cosine functions are fundamental in trigonometry.
   • These derivatives are periodic and play a key role in oscillatory motion.
   • Understanding these derivatives is crucial for applications in physics and engineering.

10. Fundamental Theorem of Calculus: ∫[a to b] f(x)dx = F(b) - F(a), where F'(x) = f(x)

• Connects differentiation and integration, two core concepts in calculus.
• It states that the definite integral of a function can be evaluated using its antiderivative.
• This theorem is foundational for understanding the area under curves and accumulation functions.