10. Laws of Derivatives

Calculus 1



Weiqi Zhou

Addition

Proof

Multiplication (The Product Rule)

Proof

The Leibniz Rule

Division

Proof

Exercise

  • complete the remaining steps

  • find the derivative of

  • find the derivative of

Linearity of Differentiation

it commutes with addition and scalar multiplication

The Chain Rule

Proof

(Let )

Exercise

  • find derivatives of following functions



Inversion

if is the inverse of ,
and both are differentiable, then

Proof

Exercise

  • find derivatives of following functions



Memorize the Table of Derivatives

Exercise

（1）find the derivative of when

（2）decide whether is differentiable at

（3）is continuous？

The Derivative of a Differentiable Function Need Not Be Continuous

Summary

  • The Rule for Addition, Multiplication, and Division

  • The Chain Rule

  • The Rule For Inversion

  • The Table