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# Calculus Rules

### Differentiation & Integration Rules

Question | Answer |
---|---|

What is the integral of 1/x dx? | ln|x| + C |

What is the integral of e^x dx? | e^x + C |

What is the integral of x^5 dx? | (x^6)/6 + C |

What is the derivative of ln(x)? | 1/x |

What is the derivative of x^5? | 5x^4 |

What is the derivative of logx? | 1/(xln10) |

What is the derivative of (3x + 7)^5? | 15(3x + 7)^4 |

What is the derivative of xln(x)? | ln(x) + 1 |

What is the derivative of (x + 5)/x? | -5/x^2 |

What is the derivative of (x^2 + 5)/x? | 1 - 5/x^2 |

What is the derivative of x/(x^2 - 1)? | (-x^2 - 1)/(x^2 - 1)^2 |

What is the derivative of sin(x)? | cos(x) |

What is the derivative of cos(x)? | -sin(x) |

What is the integral of 2 dx? | 2x + C |

What is the integral of (x + 1)/x dx? | x + ln|x| + C |

What is the integral of x/(x + 1) dx? | x + 1 - ln|x + 1| + C |

What is the integral of sinx dx? | -cosx + C |

What is the integral of cosx dx? | sinx + C |

What is the integral of (3x + 2)^7 dx? | ((3x + 2)^8)/24 |

What is the integral of 7^x dx? | (7^x)/ln7 + C |

How can the rate of change be found? | Derive! |

How can the area under a curve be found? | Integrate! |

What is the integral of (x^5 + 4x^3)/x? | (x^5)/5 + (4x^3)/3 + C |

What is the derivative of (9x^2 + 1)^4? | 72x(9x^2 + 1)^3 |

How can the average value of f(x) = x^3 - 4 between x = 2 and x = 5 be found? | 1/(5 - 2) * integral of x^3 - 4 from 2 to 5 using the Fundamental Theorem of Calculus |

Solve ln(2 + x) = 5 for x. | e^5 - 2 |

Solve log(3x - 1) = 7 for x. | (10^7 + 1)/3 or 10000007/3 |

What is the inverse of y = 2x - 3? | y = (x + 3)/2 |

What test is used to determine if an equation is a function? | Vertical line test |

What test is used to determine if a function has an inverse? | Horizontal line test |

What is the domain of lnx? | Positive real numbers |

What is the range of logx? | All real numbers |

If f(x) = 3x - 1, what is k if (k, 2) is on the inverse of f(x)? | 5 |

Solve 5lnx = 2 | e^(2/5) |

What is ln1? | 0 |

What is log1? | 0 |

When using logarithmic differentiation as a tool, what type of logarithm is used? | Natural logarithms! (The derivative of lnu = u'/u) |

What is loga + 2logb simplified? | log(a*b^2) |

What is the value of "e" rounded to the nearest whole number? | 3 |

How many derivatives must be taken of a polynomial with a degree of 23 before it equals 0? | 24 |