Differentiation Methods and Rules Study Notes

Differentiation Basics

Fundamental Differentiation Rules

  1. Constant Rule: The derivative of a constant is zero.
       - ddx(C)=0\frac{d}{dx}(C) = 0 where $C$ is a constant.

  2. Natural Logarithm Rule: The derivative of the natural logarithm of $x$ is given by:
       - ddx(ln⁡x)=1x\frac{d}{dx}(\ln{x}) = \frac{1}{x}

  3. Power Rule: If $n$ is a constant, then the derivative of $x^n$ is:
       - ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}

  4. Trigonometric Functions:
       - Derivative of sine: ddx(sin⁡x)=cos⁡x\frac{d}{dx}(\sin{x}) = \cos{x}
       - Derivative of cosine: ddx(cos⁡x)=−sin⁡x\frac{d}{dx}(\cos{x}) = -\sin{x}
       - Derivative of tangent: ddx(tan⁡x)=sec⁡2x\frac{d}{dx}(\tan{x}) = \sec^2{x}
       - Derivative of secant: ddx(sec⁡x)=sec⁡xtan⁡x\frac{d}{dx}(\sec{x}) = \sec{x} \tan{x}
       - Derivative of cosecant: ddx(csc⁡x)=−csc⁡xcot⁡x\frac{d}{dx}(\csc{x}) = -\csc{x} \cot{x}
       - Derivative of cotangent: ddx(cot⁡x)=−csc⁡2x\frac{d}{dx}(\cot{x}) = -\csc^2{x}

  5. Exponential Functions:
       - The derivative of $e^x$ is: ddx(ex)=ex\frac{d}{dx}(e^x) = e^x
       - The derivative of $a^x$ (where $a$ is a constant) is: ddx(ax)=axln⁡a\frac{d}{dx}(a^x) = a^x \ln{a}

  6. Logarithm Base $a$:
       - The derivative is given by: ddx(log⁡ax)=1xln⁡a\frac{d}{dx}(\log_a{x}) = \frac{1}{x \ln{a}}

Chain Rule

  • The chain rule states that:
    ddx(f(g(x)))=f′(g(x))g′(x)\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)

Product Rule

  • For two functions $U$ and $V$, the derivative is:
       - ddx(UV)=U′V+UV′\frac{d}{dx}(UV) = U'V + UV'

Quotient Rule

  • For two functions $U$ and $V$, the derivative is:
       - ddx(UV)=U′V−UV′V2\frac{d}{dx}\left(\frac{U}{V}\right) = \frac{U'V - UV'}{V^2}

Other Notable Derivatives

  • ddx(sin⁡3x)=3sin⁡2xcos⁡x\frac{d}{dx}(\sin^3{x}) = 3\sin^2{x}\cos{x} (using chain rule)
  • ddx(x2)=2x\frac{d}{dx}(x^2) = 2x

Implicit Differentiation

  1. If a function is defined implicitly, derivatives can still be found using the chain rule as needed. For example, differentiating both sides of an equation with respect to $x$.

Special Cases

  1. The differentials of inverse trigonometric functions include:
       - ddx(arcsin⁡x)=11−x2\frac{d}{dx}(\arcsin{x}) = \frac{1}{\sqrt{1-x^2}}
       - ddx(arctan⁡x)=11+x2\frac{d}{dx}(\arctan{x}) = \frac{1}{1+x^2}
       - ddx(arccos⁡x)=−11−x2\frac{d}{dx}(\arccos{x}) = -\frac{1}{\sqrt{1-x^2}}

Summary of Derivative Definitions

  • It is crucial to memorize these key derivatives as they form the foundation for further calculus applications.
  • Understanding their derivations helps apply them accurately in complex problems.