1 / 16

Section 2.5 – Implicit Differentiation

Section 2.5 – Implicit Differentiation. Explicit Equations. The functions that we have differentiated and handled so far can be described by expressing one variable explicitly in terms of another variable. For example: Or, in general, y = f ( x ) . Implicit Equations.

damara
Download Presentation

Section 2.5 – Implicit Differentiation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Section 2.5 – Implicit Differentiation

  2. Explicit Equations The functions that we have differentiated and handled so far can be described by expressing one variable explicitly in terms of another variable. For example: Or, in general, y = f(x).

  3. Implicit Equations Can we take the derivative of these functions? It is possible to solve some Implicit Equations for y, then differentiate: Yet, it is difficult to rewrite most Implicit Equations explicitly. Thus, we must be introduced to a new technique to differentiate these implicit functions. Some functions, however, are defined implicitly ( not in the form y = f(x) ) by a relation between x and y such as:

  4. White Board Challenge Solve for y:

  5. *Reminder* Technically the Chain Rule can be applied to every derivative:

  6. Derivatives Involving the Dependent Variable (y) The Chain Rule is Required. a. b. The derivative of y with respect to x is… the derivative of y. This is another way to write y prime. Find the derivative of each expression

  7. Instructions for Implicit Differentiation If y is an equation defined implicitly as a differentiable function of x, to find the derivative: • Differentiate both sides of the equation with respect to x. (Remember that y is really a function of x for part of the curve and use the Chain Rule when differentiating terms containing y) • Collect all terms involving dy/dxon the left side of the equation, and move the other terms to the right side. • Factor dy/dxout of the left side • Solve for dy/dx

  8. Example 1 Differentiate both sides. Product AND Constant Multiple Rules Chain Rule Solve for dy/dx If is a differentiable function of x such that find .

  9. Example 2 Differentiate both sides Product Rule Chain Rule Twice Find if .

  10. Example 3 Find the first derivative by Differentiating both sides. Quotient Rule Chain Rule Remember: Remember: Find if . Now Find the Second Derivative

  11. Example 4 Find the derivative by differentiating both sides. Chain Rule Evaluate the derivative at x=5 and y=4. Find the slope of a line tangent to the circle at the point .

  12. White Board Challenge Find the derivative of:

  13. Example 5 Find the derivative by differentiating both sides. Chain Rule If and , find .

  14. Example 5 (continued) Evaluate the derivative with the given information. If and , find .

  15. Example 6 Find an equation of the tangent to the circle at the point . Now evaluate the derivative at x=3 and y=4. Find the derivative by differentiating both sides. Chain Rule Use the Point-Slope Formula to find the equation of the tangent line

  16. White Board Challenge Find the second derivative of:

More Related