1

Sinusoidal functions

𝑑sin (𝜃 )𝑑𝜃 |

𝜃=𝜃0=cos (𝜃0 )

𝑑cos (𝜃 )𝑑𝜃 |

𝜃=𝜃0=−sin (𝜃0 )

Products andquotientsChain rule

∆ 𝑓

∆ 𝑥

Derivatives and differentials

Differentiation

𝑑 𝑓𝑑 𝑥 |𝑥=𝑥0

≔ lim∆ 𝑥→ 0

𝑓 (𝑥0+∆ 𝑥 )− 𝑓 (𝑥0 )∆𝑥

𝑑𝑥𝑛

𝑑𝑥 |𝑥=𝑥0

=𝑛𝑥0𝑛−1

Power rule

𝑓 (𝑥 )

Slope and derivative

2

𝑥0 𝑥0+∆ 𝑥

𝑓 (𝑥0+∆ 𝑥 )

𝑥0

𝑓 (𝑥0 )

∆ 𝑓

∆ 𝑥

riserun

= Δ 𝑓Δ𝑥|

𝑥=𝑥0≔

𝑓 (𝑥0+∆ 𝑥 )− 𝑓 (𝑥0 )∆𝑥

𝑑 𝑓𝑑 𝑥 |𝑥=𝑥0

≔ lim∆ 𝑥→ 0

𝑓 (𝑥0+∆ 𝑥 )− 𝑓 (𝑥0 )∆𝑥

∫ stuff 𝑑 𝑓Differentials are not numbers

3

∆ 𝑓

∆ 𝑥• Paired appearances of df and dx, such as in , refer to the shrinkage of Dx

with the corresponding amount of shrinkage in Df. Think of differentials as shorthand instructions for limiting processes rather than as numbers.

• When you find an equation with a differential df, can you find its partner? Can you draw a triangle having legs with lengths Df and Dx?

• Regardless of how closely we look in this course, we find no tick marks on the number line corresponding to df or dx. Differentials are not numbers.

#

0

+1

+2

+3

+4

+5

-1

-2

-3

-4

-5

𝑑 𝑓𝑑𝑥

𝑓 (𝑥 )

𝑥0

Weierstrass function

4

5

Sinusoidal functions

𝑑sin (𝜃 )𝑑𝜃 |

𝜃=𝜃0=cos (𝜃0 )

𝑑cos (𝜃 )𝑑𝜃 |

𝜃=𝜃0=−sin (𝜃0 )

Chain rule

∆ 𝑓

∆ 𝑥

Derivatives and differentials

Differentiation

𝑑 𝑓𝑑 𝑥 |𝑥=𝑥0

≔ lim∆ 𝑥→ 0

𝑓 (𝑥0+∆ 𝑥 )− 𝑓 (𝑥0 )∆𝑥

𝑑𝑥𝑛

𝑑𝑥 |𝑥=𝑥0

=𝑛𝑥0𝑛−1

Power rule Products andquotients

𝑑 𝑓𝑑 𝑥 |

𝑥=𝑥0= lim∆ 𝑥→ 0

𝑎𝑥02+2𝑎𝑥0∆𝑥+𝑎∆ 𝑥2−𝑎𝑥02

∆ 𝑥

Derivative of a quadratic function

6

𝑥0

𝑓 (𝑥 )𝑓 (𝑥 )=𝑎𝑥2

𝑥0+∆ 𝑥

𝑑 𝑓𝑑 𝑥 |

𝑥=𝑥0≔ lim

∆ 𝑥→ 0

𝑓 (𝑥0+∆ 𝑥 )− 𝑓 (𝑥0 )∆𝑥

𝑑 𝑓𝑑 𝑥 |

𝑥=𝑥0= lim∆ 𝑥→0

𝑎 (𝑥0+∆𝑥 )2−𝑎𝑥02

∆ 𝑥

𝑑 𝑓𝑑 𝑥 |

𝑥=𝑥0=2𝑎𝑥0

𝑑 𝑓𝑑 𝑥 |

𝑥=𝑥0= lim∆ 𝑥→0

2𝑎𝑥0∆𝑥+𝑎∆ 𝑥2

∆𝑥𝑑 𝑓𝑑 𝑥 |

𝑥=𝑥0= lim∆ 𝑥→0

(2𝑎𝑥0+𝑎∆ 𝑥 )𝑥0

¿ lim∆ 𝑥→ 0

∑𝑘=0

𝑛 𝑛!(𝑛−𝑘) !𝑘 !

𝑥0𝑛−𝑘∆ 𝑥𝑘−𝑥0

𝑛

∆ 𝑥

Derivative of power law

7

𝑥0

𝑓 (𝑥 )𝑓 (𝑥 )=𝑥𝑛

𝑥0+∆ 𝑥

𝑑 𝑓𝑑 𝑥 |

𝑥=𝑥0≔ lim

∆ 𝑥→ 0

𝑓 (𝑥0+∆ 𝑥 )− 𝑓 (𝑥0 )∆𝑥

𝑑 𝑓𝑑 𝑥 |

𝑥=𝑥0= lim∆ 𝑥→0

(𝑥0+∆ 𝑥 )𝑛−𝑥0𝑛

∆ 𝑥

𝑑 𝑓𝑑 𝑥 |

𝑥=𝑥0=𝑛𝑥0

𝑛− 1

𝑥0

¿ lim∆ 𝑥→ 0

𝑛 !(𝑛−0 )! 0 !

𝑥0𝑛+ 𝑛 !

(𝑛−1 )!1 !𝑥0𝑛− 1∆ 𝑥+∑

𝑘=2

𝑛 𝑛 !(𝑛−𝑘 )!𝑘!

𝑥0𝑛−𝑘∆𝑥𝑘− 𝑥0

𝑛

∆ 𝑥

¿ lim∆ 𝑥→ 0

∑𝑘=0

𝑛 𝑛!(𝑛−𝑘) !𝑘 !

𝑥0𝑛−𝑘∆ 𝑥𝑘−𝑥0

𝑛

∆ 𝑥

Derivative of power law

8

𝑓 (𝑥 )=𝑥𝑛

𝑑 𝑓𝑑 𝑥 |

𝑥=𝑥0≔ lim

∆ 𝑥→ 0

𝑓 (𝑥0+∆ 𝑥 )− 𝑓 (𝑥0 )∆𝑥

𝑑 𝑓𝑑 𝑥 |

𝑥=𝑥0=𝑛𝑥0

𝑛− 1

¿ lim∆ 𝑥→ 0 ( 𝑛 !

(𝑛−1 ) !1!𝑥0𝑛−1+∑

𝑘=2

𝑛 𝑛 !(𝑛−𝑘 ) !𝑘!

𝑥0𝑛−𝑘∆𝑥𝑘−1)

¿ 𝑛 !(𝑛−1 )! 1!

𝑥0𝑛− 1=

𝑛 ∙ (𝑛−1 ) ∙ (𝑛−2 )⋯3 ∙2 ∙1(𝑛−1 ) ∙ (𝑛−2 )⋯ 3 ∙2 ∙1 ∙1

𝑥0𝑛− 1

𝑥

0

𝑓 (𝑥 )

𝑥0+∆ 𝑥𝑥0

stuff∆ 𝑥+s

tuff ∆𝑥2 +st

uff ∆𝑥3 +⋯

9

Sinusoidal functions

𝑑sin (𝜃 )𝑑𝜃 |

𝜃=𝜃0=cos (𝜃0 )

𝑑cos (𝜃 )𝑑𝜃 |

𝜃=𝜃0=−sin (𝜃0 )

∆ 𝑓

∆ 𝑥

Derivatives and differentials

Differentiation

𝑑 𝑓𝑑 𝑥 |𝑥=𝑥0

≔ lim∆ 𝑥→ 0

𝑓 (𝑥0+∆ 𝑥 )− 𝑓 (𝑥0 )∆𝑥

𝑑𝑥𝑛

𝑑𝑥 |𝑥=𝑥0

=𝑛𝑥0𝑛−1

Power rule Products andquotientsChain rule

Chain rule

10

𝑓 (𝑥 )

𝑑 𝑓𝑑 𝑥 |

𝑥=𝑥0≔ lim

∆ 𝑥→ 0

𝑓 (𝑥0+∆ 𝑥 )− 𝑓 (𝑥0 )∆𝑥

𝑥0

𝑦= 𝑓 (𝑥 )

𝑧=𝑔 (𝑦 )

𝑔 ( 𝑦 )

𝑔 ( 𝑓 (𝑥 ) )

𝑑𝑔 ( 𝑓 (𝑥 ) )𝑑𝑥 |

𝑥=𝑥0=𝑑𝑔 ( 𝑓 (𝑥 ) )𝑑 𝑓 (𝑥 ) |𝑓 (𝑥 )= 𝑓 (𝑥0 )

𝑑 𝑓 (𝑥 )𝑑𝑥 |

𝑥=𝑥0

∆ 𝑥∆ 𝑓

∆ 𝑓∆ 𝑔

∆ 𝑔∆ 𝑥

∆ 𝑓 ≅ 𝑑 𝑓 (𝑥 )𝑑𝑥 |

𝑥=𝑥0∆𝑥

∆ 𝑔≅ 𝑑𝑔 ( 𝑓 (𝑥 ) )𝑑 𝑓 (𝑥 ) |𝑓 (𝑥 ) = 𝑓 (𝑥0 )

∆ 𝑓

∆ 𝑔≅ 𝑑𝑔 ( 𝑓 (𝑥 ) )𝑑 𝑓 (𝑥 ) |𝑓 (𝑥 ) = 𝑓 (𝑥0 )

𝑑 𝑓 (𝑥 )𝑑𝑥 |

𝑥=𝑥0∆𝑥

∆𝑔∆ 𝑥 ≅

𝑑𝑔 ( 𝑓 (𝑥 ) )𝑑 𝑓 (𝑥 ) |𝑓 (𝑥 )= 𝑓 (𝑥0 )

𝑑 𝑓 (𝑥 )𝑑𝑥 |

𝑥=𝑥0

11

Sinusoidal functions

𝑑sin (𝜃 )𝑑𝜃 |

𝜃=𝜃0=cos (𝜃0 )

𝑑cos (𝜃 )𝑑𝜃 |

𝜃=𝜃0=−sin (𝜃0 )

∆ 𝑓

∆ 𝑥

Derivatives and differentials

Differentiation

𝑑 𝑓𝑑 𝑥 |𝑥=𝑥0

≔ lim∆ 𝑥→ 0

𝑓 (𝑥0+∆ 𝑥 )− 𝑓 (𝑥0 )∆𝑥

𝑑𝑥𝑛

𝑑𝑥 |𝑥=𝑥0

=𝑛𝑥0𝑛−1

Power rule Products andquotientsChain rule

𝑥0+∆ 𝑥

𝑥0

Product rule

12

𝑥

0 𝑓 (𝑥 )

𝑔 (𝑥 )

𝑓 (𝑥0+∆ 𝑥 )𝑓 (𝑥0 )

𝑔(𝑥

0+∆𝑥 )

𝑔(𝑥

0)

𝐴 (𝑥 )= 𝑓 (𝑥 )𝑔 (𝑥 ) 𝑑 𝐴𝑑𝑥 |

𝑥=𝑥0≔ lim

∆𝑥→ 0

𝐴 (𝑥0+∆ 𝑥 )−𝐴 (𝑥0 )∆ 𝑥∆ 𝐴=𝐴 (𝑥0+∆𝑥 )−𝐴 (𝑥0 )

Product rule

13

𝑥0+∆ 𝑥

𝑥0

𝑥

0 𝑓 (𝑥 )

𝑔 (𝑥 )

𝑑 𝐴𝑑𝑥 |

𝑥=𝑥0≔ lim

∆𝑥→ 0

𝐴 (𝑥0+∆ 𝑥 )−𝐴 (𝑥0 )∆ 𝑥

𝐴 (𝑥 )= 𝑓 (𝑥 )𝑔 (𝑥 )∆ 𝐴=𝐴 (𝑥0+∆𝑥 )−𝐴 (𝑥0 )

Δ 𝐴Δ𝑥 =

𝑓 (𝑥0+∆ 𝑥 )− 𝑓 (𝑥0 )∆𝑥 𝑔 (𝑥0 )+ 𝑓 (𝑥0 )

𝑔 (𝑥0+∆ 𝑥 )−𝑔 (𝑥0 )∆ 𝑥

𝑑 𝐴𝑑𝑥 |

𝑥=𝑥0= 𝑑 𝑓𝑑 𝑥|

𝑥=𝑥0𝑔 (𝑥0 )+ 𝑓 (𝑥0 ) 𝑑𝑔𝑑𝑥 |

𝑥=𝑥0

𝑓 𝑔

𝑓 (𝑥0+∆ 𝑥 )𝑓 (𝑥0 )

𝑔(𝑥

0+∆𝑥 )

𝑔(𝑥

0)

Quotient rule

14

𝑑 𝐴𝑑𝑥 |

𝑥=𝑥0≔ lim

∆𝑥→ 0

𝐴 (𝑥0+∆ 𝑥 )−𝐴 (𝑥0 )∆ 𝑥

𝐴 (𝑥 )= 𝑓 (𝑥 )𝑔 (𝑥 )𝑑 𝐴𝑑𝑥 |

𝑥=𝑥0= 𝑑 𝑓𝑑 𝑥|

𝑥=𝑥0𝑔 (𝑥0 )+ 𝑓 (𝑥0 ) 𝑑𝑔𝑑𝑥 |

𝑥=𝑥0

STOP

𝑑𝑄𝑑𝑥 |𝑥=𝑥0

=

𝑑 𝑓𝑑 𝑥|

𝑥=𝑥0𝑔 (𝑥0 )− 𝑓 (𝑥0 ) 𝑑𝑔𝑑𝑥 |

𝑥=𝑥0

[𝑔 (𝑥0 ) ]2

𝑄 (𝑥 )= 𝑓 (𝑥 )𝑔 (𝑥 )

For the quotient , what is the derivative?

Why?

15

Sinusoidal functions

𝑑sin (𝜃 )𝑑𝜃 |

𝜃=𝜃0=cos (𝜃0 )

𝑑cos (𝜃 )𝑑𝜃 |

𝜃=𝜃0=−sin (𝜃0 )

∆ 𝑓

∆ 𝑥

Derivatives and differentials

Differentiation

𝑑 𝑓𝑑 𝑥 |𝑥=𝑥0

≔ lim∆ 𝑥→ 0

𝑓 (𝑥0+∆ 𝑥 )− 𝑓 (𝑥0 )∆𝑥

𝑑𝑥𝑛

𝑑𝑥 |𝑥=𝑥0

=𝑛𝑥0𝑛−1

Power rule Products andquotientsChain rule

Derivative of sine

16

𝑓 (𝜃 )=sin (𝜃 ) 𝑑 𝑓𝑑𝜃|

𝜃=𝜃0≔ lim

∆ 𝜃→0

𝑓 (𝜃0+∆𝜃 )− 𝑓 (𝜃0 )∆𝜃

𝑑 𝑓𝑑𝜃|

𝜃=𝜃0= lim∆𝜃→0

sin (𝜃0+∆ 𝜃 )−sin (𝜃0 )∆ 𝜃

Recall angle-addition identity from video on trigonometry

𝑑 𝑓𝑑 𝜃|

𝜃=𝜃0= lim∆𝜃→0

sin (𝜃 0 )cos (∆𝜃 )+cos (𝜃0 )sin (∆𝜃 )− sin (𝜃0 )∆𝜃

𝑑 𝑓𝑑𝜃|

𝜃=𝜃0= lim∆𝜃→0

sin (𝜃0 ) (cos (∆𝜃 )−1 )+cos (𝜃0 ) sin (∆𝜃 )∆ 𝜃

𝑑 𝑓𝑑 𝜃|

𝜃=𝜃0= lim∆𝜃→0

sin (𝜃0 )(cos (∆𝜃 )−1 )

∆ 𝜃 + lim∆ 𝜃→0

cos (𝜃0 ) sin(∆𝜃 )∆ 𝜃

0 1

𝑑 𝑓𝑑 𝜃|𝜃=𝜃0

= lim∆𝜃→0

sin (𝜃0 )(cos (∆𝜃 )−1 )

∆ 𝜃 + lim∆ 𝜃→0

cos (𝜃0 ) sin(∆𝜃 )∆ 𝜃

1

∆ 𝜃

cos (∆ 𝜃 )

∆ 𝜃

sin

(∆𝜃

)

1−cos (∆𝜃 )

17

𝑑 𝑓𝑑𝜃|

𝜃=𝜃0≔ lim

∆ 𝜃→0

𝑓 (𝜃0+∆𝜃 )− 𝑓 (𝜃0 )∆𝜃

𝑓 (𝜃 )=sin (𝜃 )

𝑑 𝑓𝑑𝜃|

𝜃=𝜃0=cos (𝜃0 )

Derivative of sine

Trigonometry: sine and cosine

18

2𝜋𝜃

0

-1

1

𝜋𝜋4

𝜋2

3𝜋2

12

−1 /2

√2/2√3 /2

−√3 /2−√2 /2

3𝜋4

5𝜋4

7𝜋4

𝜋6

𝜋3

2𝜋35𝜋6

7𝜋6

4𝜋3

5𝜋311𝜋6

sin (𝜃 )

cos (𝜃 )

𝑑sin (𝜃 )𝑑𝜃 |

𝜃=𝜃0=cos (𝜃0 )

Derivative of cosine

19

𝑑sin (𝜃 )𝑑𝜃 |

𝜃=𝜃0=cos (𝜃0 )

𝑑cos (𝜃 )𝑑𝜃 |

𝜃=𝜃0=−sin (𝜃0 )

𝜃

𝜋2 −𝜃

cos (𝜃 )=sin (𝜋2 −𝜃)

sin (𝜃 )=cos (𝜋2 −𝜃)

STOPHint:

20

Sinusoidal functions

𝑑sin (𝜃 )𝑑𝜃 |

𝜃=𝜃0=cos (𝜃0 )

𝑑cos (𝜃 )𝑑𝜃 |

𝜃=𝜃0=−sin (𝜃0 )

∆ 𝑓

∆ 𝑥

Derivatives and differentials

Differentiation

𝑑 𝑓𝑑 𝑥 |𝑥=𝑥0

≔ lim∆ 𝑥→ 0

𝑓 (𝑥0+∆ 𝑥 )− 𝑓 (𝑥0 )∆𝑥

𝑑𝑥𝑛

𝑑𝑥 |𝑥=𝑥0

=𝑛𝑥0𝑛−1

Power rule Products andquotientsChain rule