Calc 1 Formulas

Calculus Power Rules

  • Power Rule:
      ddxxn=nxn−1\frac{d}{dx} x^n = n x^{n-1}

Quotient Rule

  • Quotient Rule:
      ddxf(x)g(x)=g(x)ddxf(x)−f(x)ddxg(x)[g(x)]2\frac{d}{dx} \frac{f(x)}{g(x)} = \frac{g(x) \frac{d}{dx} f(x) - f(x) \frac{d}{dx} g(x)}{[g(x)]^2}

Chain Rule

  • Chain Rule:
      ddxf(g(x))=f′(g(x))⋅g′(x)\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)

Product Rule

  • Product Rule:
      ddx[f(x)⋅g(x)]=f(x)⋅g′(x)+f′(x)⋅g(x)\frac{d}{dx} [f(x) \cdot g(x)] = f(x) \cdot g'(x) + f'(x) \cdot g(x)

Reciprocal Rule

  • Reciprocal Rule:
      ddx1g(x)=−g′(x)[g(x)]2\frac{d}{dx} \frac{1}{g(x)} = -\frac{g'(x)}{[g(x)]^2}

Integration-by-Parts

  • Integration-by-Parts Formula:
      ∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du

Trigonometric Functions Derivatives and Integrals

  • Derivation and Integration of Trigonometric Functions:
      - For sin⁡x\sin x:
        - Derivative:
          ddxsin⁡x=cos⁡x\frac{d}{dx} \sin x = \cos x
        - Integral:
          ∫sin⁡x dx=−cos⁡x+c\int \sin x \, dx = -\cos x + c

      - For cos⁡x\cos x:
        - Derivative:
          ddxcos⁡x=−sin⁡x\frac{d}{dx} \cos x = -\sin x
        - Integral:
          ∫cos⁡x dx=sin⁡x+c\int \cos x \, dx = \sin x + c

  - For tan⁡x\tan x:
    - Derivative:
      ddxtan⁡x=sec⁡2x\frac{d}{dx} \tan x = \sec^2 x
    - Integral:
      ∫tan⁡x dx=ln⁡∣sec⁡x∣+c\int \tan x \, dx = \ln |\sec x| + c

  - For sec⁡x\sec x:
    - Derivative:
      ddxsec⁡x=sec⁡xtan⁡x\frac{d}{dx} \sec x = \sec x \tan x
    - Integral:
      ∫sec⁡x dx=ln⁡∣sec⁡x+tan⁡x∣+c\int \sec x \, dx = \ln |\sec x + \tan x| + c

  - For cot⁡x\cot x:
    - Derivative:
      ddxcot⁡x=−csc⁡2x\frac{d}{dx} \cot x = -\csc^2 x
    - Integral:
      ∫cot⁡x dx=ln⁡∣sin⁡x∣+c\int \cot x \, dx = \ln |\sin x| + c

  - For csc⁡x\csc x:
    - Derivative:
      ddxcsc⁡x=−csc⁡xcot⁡x\frac{d}{dx} \csc x = -\csc x \cot x
    - Integral:
      ∫csc⁡x dx=ln⁡∣csc⁡x−cot⁡x∣+c\int \csc x \, dx = \ln |\csc x - \cot x| + c

Inverse Trigonometric Functions Derivatives and Integrals

  • Inverse Trigonometric Functions:
      - For sin⁡−1(x)\sin^{-1}(x):
        - Derivative:
          ddxsin⁡−1(x)=11−x2\frac{d}{dx} \sin^{-1}(x) = \frac{1}{\sqrt{1 - x^2}}
        - Integral:
          ∫sin⁡−1(x) dx=xsin⁡−1(x)+1−x2+c\int \sin^{-1}(x) \, dx = x \sin^{-1}(x) + \sqrt{1 - x^2} + c

  - For cos⁡−1(x)\cos^{-1}(x):
    - Derivative:
      ddxcos⁡−1(x)=−11−x2\frac{d}{dx} \cos^{-1}(x) = -\frac{1}{\sqrt{1 - x^2}}
    - Integral:
      ∫cos⁡−1(x) dx=xcos⁡−1(x)+1−x2+c\int \cos^{-1}(x) \, dx = x \cos^{-1}(x) + \sqrt{1 - x^2} + c

  - For tan⁡−1(x)\tan^{-1}(x):
    - Derivative:
      ddxtan⁡−1(x)=11+x2\frac{d}{dx} \tan^{-1}(x) = \frac{1}{1 + x^2}
    - Integral:
      ∫tan⁡−1(x) dx=xtan⁡−1(x)−12ln⁡(1+x2)+c\int \tan^{-1}(x) \, dx = x \tan^{-1}(x) - \frac{1}{2} \ln(1 + x^2) + c

Identities of Trigonometric Functions

  • Trigonometric Identities:
      - sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1
      - sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2 \sin x \cos x
      - 1+cot⁡2x=csc⁡2x1 + \cot^2 x = \csc^2 x
      - cos⁡2x=cos⁡2x−sin⁡2x\cos 2x = \cos^2 x - \sin^2 x
      - cos⁡2x=1+cos⁡2x2\cos^2 x = \frac{1 + \cos 2x}{2}
      - sin⁡x=1−cos⁡2x2\sin x = \frac{1 - \cos 2x}{2}
      - tan⁡2x=2tan⁡x1−tan⁡2x\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}

Exponential Functions Derivatives and Integrals

  • Exponential Functions:
      - For exe^x:
        - Derivative:
          ddxex=ex\frac{d}{dx} e^x = e^x
        - Integral:
          ∫ex dx=ex+c\int e^x \, dx = e^x + c

      - For bxb^x:
        - Derivative:
          ddxbx=ln⁡(b)bx\frac{d}{dx} b^x = \ln(b) b^x
        - Integral:
          ∫bx dx=bxln⁡b+c\int b^x \, dx = \frac{b^x}{\ln b} + c

Logarithmic Functions Derivatives and Integrals

  • Logarithmic Functions:
      - For natural logarithm ln⁡(x)\ln(x):
        - Derivative:
          ddxln⁡(x)=1x\frac{d}{dx} \ln(x) = \frac{1}{x}
        - Integral:
          ∫ln⁡(x) dx=xln⁡(x)−x+c\int \ln(x) \, dx = x \ln(x) - x + c

      - For logarithm base b log⁡b(x)\log_b(x):
        - Derivative:
          ddxlog⁡b(x)=1xln⁡(b)\frac{d}{dx} \log_b(x) = \frac{1}{x \ln(b)}
        - Integral:
          ∫log⁡b(x) dx=xln⁡(x)−xln⁡(b)+c\int \log_b(x) \, dx = \frac{x \ln(x) - x}{\ln(b)} + c

Change of Base Formula

  • Change of Base Formula for Logarithms:
      log⁡b(x)=ln⁡(x)ln⁡(b)\log_b(x) = \frac{\ln(x)}{\ln(b)}

Definitions of Logs

  • Definition of Logarithm Base b:
      log⁡b(N)=x iff bx=N\log_b(N) = x \text{ iff } b^x = N

  • Special Values:
      - ln⁡(e)=1\ln(e) = 1
      - log⁡b(1)=0\log_b(1) = 0
      - log⁡b(b)=1\log_b(b) = 1

Special Identities

  • ln⁡(1)=0\ln(1) = 0

  • blog⁡b(x)=xb^{\log_b(x)} = x

  • eln⁡(x)=xe^{\ln(x)} = x

  • log⁡b(xy)=ylog⁡b(x)\log_b(x^y) = y \log_b(x)

  • ln⁡(ab)=ln⁡(a)+ln⁡(b)\ln(ab) = \ln(a) + \ln(b)

These notes provide an extensive overview of critical calculus concepts and their mathematical representations, aimed at strengthening understanding and application in both theoretical and practical scenarios.