1. Implicit Differentiation

2. Determine the gradient of xy + y  4x =  2 at the point (5 , 3).
Explicit Differentiation
Explicit Differentiation
Explicitly Defined
(y isolated and all other terms on the other side without any y’s)

3. Determine the gradient of xy + y  4x =  2 at the point (5 , 3).
Implicit Differentiation
Before we begin we need to be able to find the derivative of y with respect to x
By Chain Rule
Another Example

4. Determine the gradient of xy + y  4x =  2 at the point (5 , 3).
Implicit Differentiation
Differentiate with respect to x.
By Product Rule

5. Determine the gradient of xy + y  4x =  2 at the point (5 , 3).
Implicit Differentiation
Differentiate with respect to x.
Same Result as before 

6. If x2 + y2 = 36 find dy/dx when x=2.
Differentiate implicitly with respect to x.
Top Point
Bottom Point

7. Find the equations of the lines tangent and normal to the curve at .
We need the gradient. Since we can’t solve for y, we use implicit differentiation to solve for
Using the product rule.

8. Find the equations of the lines tangent and normal to the curve at .

9. Implicit Differentiation and Higher Order Derivatives
Find if
Substitute
back into the equation.