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Derivatives Cheatsheet

Derivatives

Derivative Definition

\(\frac{d}{dx}(f(x)) = f'(x) = \lim_{ h \to 0 }\frac{f(x+h)-f(x)}{h}\)

\((cf(x))' = c(f'(x))\)

\((f(x)\pm g(x))' = f'(x)g(x) \pm f(x)g'(x)\)

\(\frac{d}{dx}(c) = 0\)

Mean Value Theorem

If f is differentiable on the interval (a, b) and continuous at the endpoints, there exists a c in (a, b) s.t.

\(f'(c) = \frac{f(b)-f(a)}{b-a}\)

Product Rule

\((f(x)g(x))' = f'(x)g(x) + f(x)g'(x)\)

Quotient Rule

\(\left( \frac{f(x)}{g(x)} \right)' = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}\)

Power Rule

\(\frac{d}{dx}(x^n) = nx^{n-1}\)

Chain Rule

\(\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)\)

Common Derivatives

  1. Constant Function: \(\frac{d}{dx}(c) = 0\)
  2. Power of \(x\) : \(\frac{d}{dx}(x^n) = nx^{n-1}\)
  3. Trigonometric Functions:
  4. \(\frac{d}{dx}(\sin(x)) = \cos(x)\)
  5. \(\frac{d}{dx}(\cos(x)) = -\sin(x)\)
  6. \(\frac{d}{dx}(\tan(x)) = \sec^2(x)\)
  7. \(\frac{d}{dx}(\sec(x)) = \sec(x)\tan(x)\)
  8. \(\frac{d}{dx}(\csc(x)) = -\csc(x)\cot(x)\)
  9. \(\frac{d}{dx}(\cot(x)) = -\csc^2(x)\)
  10. Inverse Trigonometric Functions:
  11. \(\frac{d}{dx}(\arcsin(x)) = \frac{1}{\sqrt{1 - x^2}}\)
  12. \(\frac{d}{dx}(\arccos(x)) = \frac{-1}{\sqrt{1 - x^2}}\)
  13. \(\frac{d}{dx}(\arctan(x)) = \frac{1}{1 + x^2}\)
  14. Exponential/Logarithmic Functions
  15. \(\frac{d}{dx}(a^{x)}= a^x\ln a\)
  16. \(\frac{d}{dx}(e^{x})= e^x\)
  17. \(\frac{d}{dx}(\ln x) = \frac{1}{x}, x>0\)
  18. \(\frac{d}{dx}(\ln(|x|)) = \frac{1}{x}\)
  19. \(\frac{d}{dx}(\log_{a}(x)) = \frac{1}{x\ln a}\)

Integration

Integral Definiton

\(\int_{a}^{b} \,f(x) dx = \lim_{ n \to \infty } \Sigma^{n}_{i =1}f(x_{i}^{*}) \Delta x\)

Fundamental Theorem of Calculus Pt. 1

If \(f\) continuous on \([a, b]\) then

\(g(x) = \int^{x}_{a}\,f(t)dt\) is also continuous on \([a, b]\) and

\(g'(x) = \frac{d}{dx}\int ^{x}_{a}\,f(t)dt\).

Fundamental Theorem of Calculus Pt. 2

\(f\) is continuous on \([a, b]\), \(F(x)\) is an anti-derivative of \(f(x)\) i.e. \(F(x) = \int \,f(x) dx\), then

\(\int^{b}_{a} \,f(x)dx = F(b) - F(a)\)

Properties

Certainly! Below is a section on the properties of integrals:

Properties of Integrals

  1. Linearity:
  2. Scalar Multiplication: \(\int_a^b c \cdot f(x) \, dx = c \cdot \int_a^b f(x) \, dx\)

  3. Addition/Subtraction: \(\int_a^b (f(x) \pm g(x)) \, dx = \int_a^b f(x) \, dx \pm \int_a^b g(x) \, dx\)

  4. Interval Properties:

  5. Reversal of Limits: \(\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx\)
  6. Zero Width: \(\int_a^a f(x) \, dx = 0\)

  7. Additivity: \(\int_a^b f(x) \, dx + \int_b^c f(x) \, dx = \int_a^c f(x) \, dx\)

  8. Comparison: If \(f(x) \geq g(x)\) for all \(x\) in \([a, b]\), then \(\int_a^b f(x) \, dx \geq \int_a^b g(x) \, dx\).

  9. Absolute Values: \(\left| \int_a^b f(x) \, dx \right| \leq \int_a^b |f(x)| \, dx\)

  10. Mean Value Theorem for Integrals: If \(f\) is continuous on \([a, b]\), then there exists some \(c\) in \((a, b)\) such that \(\int_a^b f(x) \, dx = f(c)(b - a)\).

Certainly! Here's a list of common integrals:

Common Integrals

  1. Power Rule (for \(n \neq -1\)):

\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)

  1. Exponential Function:

\(\int e^x \, dx = e^x + C\)

  1. Natural Logarithm:

\(\int \frac{1}{x} \, dx = \ln|x| + C\)

  1. Trigonometric Functions:
  2. \(\int \sin(x) \, dx = -\cos(x) + C\)
  3. \(\int \cos(x) \, dx = \sin(x) + C\)
  4. \(\int \tan(x) \, dx = -\ln|\cos(x)| + C\) or \(-\ln|\sec(x)| + C\)
  5. \(\int \sec^2(x) \, dx = \tan(x) + C\)
  6. \(\int \sec(x)\tan(x) \, dx = \sec(x) + C\)
  7. \(\int \csc^2(x) \, dx = -\cot(x) + C\)

  8. \(\int \csc(x)\cot(x) \, dx = -\csc(x) + C\)

  9. Inverse Trigonometric Functions:

  10. \(\int \arcsin(x) \, dx = x \arcsin(x) + \sqrt{1-x^2} + C\)
  11. \(\int \arccos(x) \, dx = x \arccos(x) - \sqrt{1-x^2} + C\)

  12. \(\int \arctan(x) \, dx = x \arctan(x) - \frac{1}{2} \ln |1+x^2| + C\)

  13. Exponential Growth/Decay:

\(\int a \cdot e^{kx} \, dx = \frac{a}{k} \cdot e^{kx} + C\)

  1. Logarithmic Functions (for \(a > 0\), \(a \neq 1\)):

\(\int \log_a(x) \, dx = x (\ln(x) \log_a(e) - \ln(a) \log_a(e)) + C\)

In all of the above, \(C\) represents the constant of integration. Remember, Indefinite Integrals are all ==sets of functions== not specific functions.