Differentiation Formulas in Calculus Lecture Notes

Derivative of a Constant

• The derivative of a constant is always zero.

Power Rule

• Formula:
  • For a variable x raised to a constant n:
    • d/dx(x^n) = n * x^(n-1)
  • Examples:
    • d/dx(x^3) = 3x^2
    • d/dx(x^4) = 4x^3
    • d/dx(x^5) = 5x^4

Derivative of a Constant Raised to a Variable

• Formula:
  • For a constant a raised to a variable x:
    • d/dx(a^x) = a^x * ln(a)
  • For a constant a raised to a function U(x):
    • d/dx(a^U) = a^U * U' * ln(a)

Logarithmic Differentiation

• Used for finding the derivative of a variable raised to a variable.
• Further resources available on YouTube (Organic Chemistry Tutor).

Constant Multiple Rule

• Formula:
  • For a function f(x) multiplied by a constant C:
    • d/dx(C * f(x)) = C * d/dx(f(x))
  • Example:
    • d/dx(5x^4) = 5 * d/dx(x^4) = 5 * 4x^3 = 20x^3

Product Rule

• Formula:
  • For two functions u and v:
    • d/dx(u * v) = u' * v + u * v'

Quotient Rule

• Formula:
  • For a fraction of two functions u and v:
    • d/dx(u/v) = (v * u' - u * v') / v^2

Chain Rule

• Formula:
  • For a composite function f(g(x)):
    • d/dx(f(g(x))) = f'(g(x)) * g'(x)
  • For a composite function f(g(U(x))):
    • d/dx(f(g(U(x)))) = f'(g(U(x))) * g'(U(x)) * U'

Chain Rule with Power Rule

• Formula:
  • For a function f(x) raised to a constant n:
    • d/dx(f(x)^n) = n * f(x)^(n-1) * f'(x)
  • Example:
    • For any function of x: d/dx(n * f(x)^(n-1)) * f'(x)

Logarithmic Functions

• Formula:
  • For log base a of U(x):
    • d/dx(log_a(U)) = U' / (U * ln(a))
  • For natural log of U(x):
    • d/dx(ln(U)) = U' / U

Trigonometric Functions

• Formulas:
  • d/dx(sin(U)) = cos(U) * U'
  • d/dx(cos(U)) = -sin(U) * U'
  • d/dx(tan(U)) = sec^2(U) * U'
  • d/dx(cot(U)) = -csc^2(U) * U'
  • d/dx(sec(U)) = sec(U) * tan(U) * U'
  • d/dx(csc(U)) = -csc(U) * cot(U) * U'

Inverse Trigonometric Functions

• Formulas:
  • d/dx(arcsin(U)) = U' / sqrt(1 - U^2)
  • d/dx(arccos(U)) = -U' / sqrt(1 - U^2)
  • d/dx(arctan(U)) = U' / (1 + U^2)
  • d/dx(arccot(U)) = -U' / (1 + U^2)
  • d/dx(arcsec(U)) = U' / (|U| * sqrt(U^2 - 1))
  • d/dx(arccsc(U)) = -U' / (|U| * sqrt(U^2 - 1))

Important Notes

• It is crucial to understand and remember these formulas for studying derivatives in calculus.
• Additional example problems and explanations available through links (not provided here).