Implicit Differentiation

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 or • Implicit functions are only implied by an equation, and may be difficult to express explicitly such as

3. Explicit Vs. Implicit cont.

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

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

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. , 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 ,given that: • 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 (

9. Implicit Curve represented by

10. Graphs of differentiable functions • Let’s represent each of these equations as differentiable functions that we can graph (if possible) • A) • B) • C) • D)

11. Determine the slope of the tangent line to the graph • Ellipse: at point (,-) • Lemniscate: at (3,1) • Find the derivative of Inverse Sinusoidal Curve: sin y = x

12. Finding a line tangent to a graph • at point ()

13. Find the Second Derivative

14. Hw • 1-19 odd, 25, 27, 29, 35-38, 43-46