11. Derivatives of Implicit Functions

Calculus 1



Weiqi Zhou

Implicit Functions

  • explicit function:
    e.g.,：,

  • implicit function:
    e.g.,

Derivatives of Implicit Functions

  • differentiate both sides of with respect to

  • keep in mind that is a function of

  • apply the chain rule and the product rule where necessary

  • keep in the result

Example

  • find the derivative of where is determined by

  • differentiate with respect to :

  • sort it to get

Exercises

  • find the derivative of at where is determined by

  • find the tangent line of at

Higher Derivatives

  • find the second derivative of where is determined by

  • differentiate with respect to :

  • do it again:

  • sort to get

Exercies

  • find the second derivative of at where is determined by

  • find the second derivative of at where is determined by

Parameterized Functions

  • hidden in the parameterization

  • example：motion of a particle，its position coordinates with respect to the time

  • example： can be rewritten as

Derivatives



  • ，

  • if admits a continuous inverse, and are differentiable, then

Exercise：why is a continuous inverse of required in the last step?

Example

  • Find the tangent line of the ellipse at





  • the slope is at the tangent point
    the line equation:

Exercise

let be the trajectory of particle

find its velocity at time , and the derivative of with respect to

Higher Derivatives



  • View as a function of

Examples and Exercises

  • find the second derivative of with respect to



  • differentiate with respect to , （Exercise）

  • （Exercise）

Summary

  • Implicit Functions

  • Derivative and Higher Derivatives of Implicit Functions

  • Parameterized Functions

  • Derivatve and Higher Derivatives of Implicit Functions