Implicit Differentiation in Functions

Implicit vs. explicit functions and learn the steps of implicit differentiation. Understand the challenges in defining implicit functions explicitly and find equations for tangent lines and derivatives. Discover the interplay between variables and rules in differentiation.

• Differentiation
• Implicit functions
• Explicit functions
• Variables
• Equations

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Presentation Transcript

1. Implicit Differentiation 3.5

2. Explicit vs. Implicit Functions Explicit functions are functions where one variable is clearly expressed as a function of another such as ? = 3?2 5 or ? ? = 2? 6 Implicit functions are only implied by an equation, and may be difficult to express explicitly such as ?2 2?3+ 4? = 2

3. Explicit Implicit Explicit Vs. Implicit cont. ? =1 ? ?? = 1 ? = ? 1 ? ???? = ? ??[1] ?? ??= ? 2 Using product rule on the left side and and then using chain rule (since y is still a function of x) 1? + ?? ??? = 0 ?? ??= 1 ? + ??? ??= 0 ?2 ?? ??= ? ? Since ? =1 ?, ?? 1 ?2 ??=

4. Differentiating with respect to x 3?2 3?2?? Variables agree Use power rule ? ???3 ? ??[?3] ?? Variables disagree Use chain rule 1 + 3?? ? ??[x + 3y] Use chain rule ?? 2xy?? Product Rule Chain Rule Simplify ? ??[??2] ??+ ?2

5. Implicit Differentiation Steps 1) Differentiate both sides of the equation with respect to x. 2) Collect all terms involving ?? the equation and move all other terms to the right side of the equation. 3) Factor ?? ?? out of the left side of the equation. 4) Solve for ?? ?? by dividing both sides of the equation by the factor on the left that does not contain ?? ?? ?? on the left side of

6. Most implicit functions can not be defined explicitly. If they can be defined explicitly, most of the time you need to restrict the domain. Ex. ?2+ ?2= 1 , the implicit equation of the unit circle defines y as a function of x only, if - 1 x 1 and one considers only non-negative (or non-positive) values for the values of the function.

7. Find the equation of the line tangent to the circle.

8. Find ?? ?3+ ?2 5? ?2= 4 1) Differentiate both sides with respect to x. 2) Collect the dy/dx terms on the left side of the equation. 3) Factor dy/dx out of the left side of the equation. 4) Solve for dy/dx by dividing by (3?2+ 2? 5) ??,given that:

9. Implicit Curve represented by ?3+ ?2 5? ?2= 4

10. Graphs of differentiable functions Let s represent each of these equations as differentiable functions that we can graph (if possible) A) ?2+ ?2= 1 B) ?2+ ?2= 0 C) ?2+ ?2= 4 D) ? + ?2= 1

11. Find the Second Derivative ?2+ ?2= 25

12. Finding a line tangent to a graph 2 2 ?2?2+ ?2= ?2 at point ( 2, 2)

13. Determine the slope of the tangent line to the graph Ellipse: ?2+ 4?2= 4 at point ( 2,-1 2) Lemniscate: 3(?2+ ?2) 2= 100xy at (3,1) Find the derivative of Inverse Sinusoidal Curve: sin y = x

14. Hw Day 1 page 172 1-19 odd, 25, 27, 29, Day 2 35-38, 43-46, 67, 69, 70

15. Implicit Differentiation Continued You had me until the Lemniscate

16. Quick Recap Implicit Equations are equations that can be messy to rewrite as functions so sometimes we differentiate first and rewrite as functions later. Mostly this is when we have x and y on the same side and y raised to some exponent

17. Still Recapping We use the derivative as an operation on each piece of the equation we can. Taking the derivative of an x value with respect to x works normally. Taking the derivative of y values with respect to x we have to use chain rule (ie write dy/dx next to everything) since y is still a function of x. Finally solve the equation for dy dx

18. Now Back to our Regularly Scheduled Lemniscate Determine the slope of the graph 3(x2+ y2)2=100xy At point (3,1)

19. Finding a Tangent Line Find the tangent line to the graph of x2(x2+ y2)= y2 At point 2 2 , 2 2 Called a Kappa Curve

20. Finding the Second Derivative of an implicit equation x2+ y2= 25 Given Find d2y dx2

21. Logarithmic Differentiation We were able to use logarithmic properties to help us when we were taking the logarithm of a messy function. If we are not already dealing with the logarithm of a function, we can take the logarithm of both sides and use implicit differentiation. y =(x 2)2 x2+1 ,x 2

22. Steps 1) Take the natural logarithm of both sides 2) Use logarithmic properties to simplify 3) Use implicit differentiation 4) Simplify and solve for the derivative with respect to x 5) Substitute your original equation for y

23. Find dy/dx for sin y = x Once we have found the derivative we can restrict the domain and doing so find the derivative for arcsin (x).

24. cosxy =x2 y Screen Shot 2013-11-24 at 2.30.30 PM.png

25. Hw Page 172 25,27, 32, 34, 35, 37, 43, 45, 67, 69, 70