(66) Differentiation Formulas - Notes

Differentiation Formulas Overview

Introduction

• Focus on derivatives in calculus.

• Prepare to take notes on essential differentiation formulas.

Basic Differentiation Rules

• Derivative of a Constant:

  • Always equals 0.

Power Rule

• Formula: If f(x) = x^n, then f'(x) = n * x^(n-1).

  • Example:

    • Derivative of x³: 3x²

    • Derivative of x⁴: 4x³

    • Derivative of x⁵: 5x⁴

Exponential Functions

• Constant Raised to a Variable: If f(x) = a^x, then f'(x) = a^x * ln(a).

• Constant Raised to a Function: If f(x) = a^u (where u is a function of x), then f'(x) = a^u * u' * ln(a).

Logarithmic Differentiation

• Used for derivatives of a variable raised to a variable.

• Refer to the video: "logarithmic differentiation organic chemistry tutor" for more details.

Product and Quotient Rules

• Constant Multiple Rule:

  • If f(x) = C * g(x), then f'(x) = C * g'(x).

  • Example: Derivative of 5x⁴ = 5 * (4x³) = 20x³.

• Product Rule:

  • If f(x) = u(x) * v(x), then f'(x) = u'v + uv'.

• Quotient Rule:

  • If f(x) = u(x) / v(x), then f'(x) = (v * u' - u * v') / v².

Chain Rule

• For composite functions: If f(g(u)), then f'(g(u)) = f'(g(u)) * g'(u).

• Represented as:

  • f(g(x)): f'(g(x)) * g'(x) (if g is a function of x).

Power Rule with Chain Rule

• For f(x)^n: f'(x) = n * f(x)^(n-1) * f'(x).

Logarithmic Functions

• Derivative of log_a(u): f'(x) = u'/ (u * ln(a)).

• Natural Logarithm:

  • ln(u): f'(x) = u'/u.

Trigonometric Functions

• Derivatives:

  • sin(u) → cos(u) * u'

  • cos(u) → -sin(u) * u'

  • tan(u) → sec²(u) * u'

  • cot(u) → -csc²(u) * u'

  • sec(u) → sec(u) * tan(u) * u'

  • csc(u) → -csc(u) * cot(u) * u'

Inverse Trigonometric Functions

• Derivatives:

  • arcsin(u) → u' / √(1 - u²)

  • arccos(u) → -u' / √(1 - u²)

  • arctan(u) → u' / (1 + u²)

  • arccot(u) → -u' / (1 + u²)

  • arcsec(u) → u' / (u * √(u² - 1))

  • arccsc(u) → -u' / (u * √(u² - 1))

Conclusion

• These formulas are essential for studying derivatives.

• Encouragement to check additional example problems in the description section.