Download Implicit Differentiation

Implicit Differentiation
*what is an implicit definition of a function
*How to differentiate implicitly
*Finding slopes of points on implicitly defined
curves
Implicit Differentiation….
1. ….Used when the function cannot be defined explicitly.
What is an explicitly defined function?
An explicit function is solved for y or solved for f(x).
An Implicit function is not solved for y
or f(x).
This is because it is difficult or impossible to do so.
Examples of each
Implicitly defined function
Explicitly defined function
Applying the Chain rule to see dy/
dx terms
Differentiation is taking place with respect to x.
– Consequences……..
1. When we differentiate terms with just xterms we take the derivative as we have
done before.
2. When we differentiate y-terms, we
must apply the chain rule because
we are assuming that y is defined
implicitly as a function of x.
Derivative
of outside
Derivative of the inside.
y is a function of x, so
the derivative of y is dy/
dx
Derivative of the
outside
Derivative of the Inside
First “D” Second
Second “D” First
Help making sense of this “y is implicitly a function of x”
We know how to do this from the chain
rule
We use g(x) to equal y when using function,
notation….so….
Implicit differentiation steps
•  1. Differentiate both sides of the equation
with respect to x (when differentiating a yterm you will have a dy/dx term)
•  2. Collect all dy/dx terms to the left side of
the equation
•  3. Factor out dy/dx
•  4. Solve for dy/dx by dividing both sides by
the factor on the left that does not have
the dy/dx term
An Example
Find dy/dx given . . .
1. Differentiate with respect to x
2. Group dy/dx terms on left side &
all other terms on the right side
3. Factor out dy/dx
4. Solve for dy/dx
For
What is the slope of the curve at (2,0)?
Slope of the curve at (2,0) equals the value of the
derivative at (2,0).
Find the tangent line at (2,0)
Point-Slope Formula
Problems…..
•  From page 141…
•  In-class (time dependent): you do
2,8,12,18 from page 141
•  Homework: do #1-19 odds page 141