You are about to erase your work on this activity. Are you sure you want to do this?
Updated Version Available
There is an updated version of this activity. If you update to the most recent version of this activity, then your current progress on this activity will be erased. Regardless, your record of completion will remain. How would you like to proceed?
Mathematical Expression Editor
We learn to compute the derivative of an implicit function.
1 Implicit Differentiation
Implicit Differentiation is used to find in situations where is not written as an
explicit function of . Some examples of equations where implicit differentiation is
necessary are:
To compute in these situations, we make the assumption that is an unspecified
function of and in most cases, we employ the chain rule with as the inside
function.
2 Warm-up Examples of Implicit Differentiation
example 1 Find where is an unspecified function of . We use the chain rule, with and . Then and we can conclude that
Here is a video of Warm-up 1
_
(problem 1) Compute
Treat as an unspecified function of , i.e., .
Use the Chain Rule with as the inside function.
example 2 Find where is an unspecified function of . We use the chain rule, with and . Then and we can conclude that
Here is a video of Warm-up 2
_
(problem 2a) Compute
Consider to be a function of .
Use the chain rule with as the inside function.
(problem 2b)
Consider to be a function of .
The derivative of with respect to x is .
Use the chain rule with as the inside function.
(problem 2c)
(problem 2d)
(problem 2e)
(problem 2f)
(problem 2g)
example 3 Find where is an unspecified function of . We use the chain rule, with and . Then and we can conclude that
Here is a video of Warm-up 3
_
(problem 3)
example 4 Find where is an unspecified function of . We use the product rule, with and . Then We conclude that
Here is a video of Warm-up 4
_
(problem 4)
Use the product rule
example 5 Find where is an unspecified function of . We use the product rule,
with and . Then and by the chain rule, We conclude that
Here is a video of Warm-up 5
_
(problem 5a)
Use the product rule and the chain rule
(problem 5b)
Use the product rule and the chain rule
We are now ready to find in situations where the formula describing a curve in the
-plane is not given in the form .
3 Examples of Implicit Differentiation
example 6 Find if . We assume that is a function of and we differentiate both sides with respect to :
We can now solve for algebraically:
Here is a video of Example 1
_
(problem 6a) Use the result of the example above to find the equation of the tangent
line to the unit circle at the point
The equation of the tangent line is
(problem 6b) Find if
Consider to be a function of .
Differentiate both sides with respect to .
Use the chain rule on with as the inside function.
The derivative of the inside is
Solve for by moving the terms with to one side and the rest of the terms to the
other side.
If then is
example 7 Find if . We assume that is a function of and we differentiate both sides with respect to :
We can now solve for algebraically:
Here is a video of Example 2
_
(problem 7a) Find if
Consider to be a function of .
Differentiate both sides with respect to .
Use the chain rule on with as the inside function.
The derivative of the inside is
Solve for by moving the terms with to one side and the rest of the terms to the
other side.
If then is
(problem 7b) Use the result of the problem above to find the equation of the tangent
line to the ellipse at the point
The equation of the tangent line is
example 8 Find if . We assume that is a function of and we differentiate both sides with respect to :
We can now solve for algebraically:
Here is a video of Example 3
_
(problem 8a) Use the result of the example above to find the equation of the tangent
line to the curve at the point
The equation of the tangent line is
(problem 8b) Find if
Consider to be a function of .
Differentiate both sides with respect to .
Use the chain rule on with as the inside function.
The derivative of the inside is
Use the product rule on the right hand side.
Solve for by moving the terms with to one side and the rest of the terms to the
other side.
If then is
example 9 Find if . We assume that is a function of and we differentiate both sides with respect to :
We can now solve for algebraically:
Here is a video of Example 4
_
(problem 9) Find if
Consider to be a function of .
Differentiate both sides with respect to .
Use the chain rule on with as the inside function.
The derivative of the inside is
Use the product rule on both terms on the left hand side and the term on the right
hand side.
Solve for by moving the terms with to one side and the rest of the terms to the
other side.
If then is
example 10 Find if . We assume that is a function of and we differentiate both sides with respect to :
We can now solve for algebraically:
Here is a video of Example 5
_
example 11 Find if . We assume that is a function of and we differentiate both sides with respect to :
We can now solve for algebraically:
Here is a video of Example 6
_
(problem 11) Find if
Consider to be a function of .
Differentiate both sides with respect to .
Use the chain rule on with as the inside function.
The derivative of the inside is by the product rule.
Use the product rule on the term.
Solve for by moving the terms with to one side and the rest of the terms to the
other side.
If then is
Here are some detailed, lecture style videos on implicit differentiation: