All the rules for derivatives (products, quotients, chain rule, and implicit differentiation) still apply for trigonometric functions.

Problem 1: Find the derivative of

y = sin 7x | u = 7x = angle |

= cos 7x (7x) = (cos 7x) (7) = 7 cos 7x |
(7 is written in front of the trigonometric
function to separate it from the angle.) |

Problem 2: Find the derivative of

y = 3x² sin²x | [this is a product (3x²) (sin²x) = uv] |

Remember sin²x = (sin x)² = u²
| |

Problem 1: Find the derivative

y = 7 sin x cos 8x | (you need the product rule) |

= 7 sin x (cos 8x) + cos 8x (7 sin x) | |

= 7 sin x (-sin 8x) (8x) + (cox 8x) (7 cos x) | |

= 7 sin x (-sin 8x) (8) + (cox 8x) (7 cos x) | |

clean it up | |

= -56 sin x sin 8x + 7 cos 8x cos x |

Problem 2: Find the derivative

Problem 3: Find the derivative

2x cos y + sin 4x = 5 | (use implicit differentiation) |

(2x cos y + sin 4x) = (5) | |

You need the product rule for the first term. | |

2x (cos y) + cos y (2x) + sin 4x = (5) | |

2x (-sin y) cos y (2) + cos 4x (4x) = 0 | |

2x (-sin y) cos y (2) + (cos 4x) (4) = 0 | |

Clean it up | |

-2x sin y 2 cos y + 4 cos 4x = 0 | |

You need by itself. | |

-2x sin y = -2 cos y - 4 cos 4x | |