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### Table of Derivatives, Integrals, & Constants

#### Given Constants

{$$\pi \approx 3.1415926535$$} {$$e \approx 2.7182818285$$}

#### Constant Formulas

{$$e = \sum_{n=1}^\infty = \frac{1}{n!} = \lim_{n \rightarrow \infty} \left( 1 + \frac{1}{n} \right)^n$$}

#### Definitions

##### Derivative

{\eqalign{ {d \over dx} f(x) &= f^\prime (x)\\ &= \lim_{\Delta x\rightarrow0} {{f(x + \Delta x) -f(x)} \over {\Delta x}} \\ &= \lim_{ h \rightarrow0} {{f(x + h) -f(x)} \over { h}} \\ &= \lim_{ x \rightarrow x_0} {{f(x_0) -f(x)} \over { x_0 - x}} }}

#### Differentiation Rules

##### Sum Rule

{$${d \over dx} (f(x) + g(x)) = {d \over dx} f(x) + {d \over dx} g(x)$$}

##### Difference Rule

{$${d \over dx} (f(x) - g(x)) = {d \over dx} f(x) - {d \over dx} g(x)$$}

##### Constant Multiplication Rule

{$${d \over dx} c(f(x)) = c ({d \over dx} f(x))$$}

##### Product Rule

{$${d \over dx}(f(x)g(x)) = g(x){d \over dx}f(x) + f(x){d \over dx}g(x)$$}

##### Quotient Rule

{$${d \over dx} \left({{f(x)} \over {g(x)}}\right) = { {g(x){d \over dx}f(x) - f(x){d \over dx}g(x)} \over {[g(x)]^2}}$$}

##### Chain Rule

{$$y = f(u) \text{ and } x = g(x) \Rightarrow {dy \over dx} = {dy \over du} \cdot {du \over dx}$$}

{$${d \over dx}(f(g(x)) = f^\prime (g(x))g^\prime (x)$$}

{$$(f \circ g)(x) = f(g(x))$$}

##### Eponentials and Logarithms

{$${d \over dx} e^x = e^x$$}

{$${d \over dx} \ln x = \left( {1} \over {x} \right)$$}

Sources:

Recommended:

Category:Category: Math Calculus

This is a student's notebook. I am not responsible if you copy it for homework, and it turns out to be wrong.

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### March 16, 2018

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