# Differential Calculus

## Derivatives and differentiation expressions.

Differential calculus is a subfield of calculus that studies the rates at which quantities change.

Expression | Derivatives |
---|---|

y = x^{n} | dy/dx = n x^{n-1} |

y = a x^{n} | dy/dx = a n x^{n-1} |

f(x) = a x^{n} | f'(x) = a n x^{n-1} |

y = e^{x} | dy/dx = e^{x} |

y = e^{a x} | dy/dx = a e^{a x} |

y = a^{x} | dy/dx = a^{x} ln(a) |

y = ln(x) | dy/dx = 1 / x |

y = sin(Θ) | dy/dΘ = cos(Θ) |

y = cos(Θ) | dy/dΘ = - sin(Θ) |

y = tan(Θ) | dy/dΘ = sec^{2}(Θ) |

y = cot(Θ) | dy/dΘ = cosec^{2}(Θ) |

y = sec(Θ) | dy/dΘ = tan(Θ) sec(Θ) = sin(Θ) / cos^{2}(Θ) |

y = cosec(Θ) | dy/dΘ = - cot(Θ) cosec(Θ) = - cos(Θ) / sin^{2}(Θ) |

y = sin^{-1}(x / a) | dy/dx = 1 / (a^{2} - x^{2})^{1/2} |

y = cos^{-1}(x / a) | dy/dx = - 1 / (a^{2} - x^{2})^{1/2} |

y = tan^{-1}(x / a) | dy/dx = a / (a^{2} + x^{2}) |

y = cot^{-1}(x / a) | dy/dx = - a / (a^{2} + x^{2}) |

y = sec^{-1}(x / a) | dy/dx = a / (x (x^{2} - a^{2})^{1/2}) |

y = cosec^{-1}(x / a) | dy/dx = - a / (x (x^{2} - a^{2})^{1/2}) |