Equation of Tangent Line: Hyperbola: Implicit Differentiation
General Contents
Tangent Line Contents
Implicit Differentiation Contents
Index
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Find the equation of the tangent line to the graph of
.
How do we find the slope of the tangent line?
We need the derivative,
How can we get that?
One way is to use implicit differentiation.
Which rule of differentiation can we use here?
The Power Rule and the Chain Rule.
Do the implicit differentiation.
Solve for
We get
Is this the slope of the tangent line at
?
No.
How do we get the slope at this point?
We need to evaluate
.
Do that.
Does this solve the problem?
No. This is the slope,
m
, we need. To get the equation of the tangent line we need the
y
-intercept,
b
, so we can state the equation of the line as
How do we find
b
?
Since the line passes through the point of tangency,
, we can substitute those coordinates and our value of
in
and solve for
b
.
Do the substitution.
becomes
Solve for
b
.
We get
.
Combine these results to get the equation of the tangent line.
Using these values of
m
and
b
, we get
Let’s do a graphical check. What is the shape of the graph of
We can divide by 400 to get
, the equation of a hyperbola.
Graph this on paper. Then check your graph by clicking on “Next”.
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Add the point
, or (6, 2.65). Then check your graph by clicking on “Next”.
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Add the tangent line to your graph. Then check by clicking “Next”.
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Does the slope appear to be correct?
Yes, it is positive and about equal to
, compared to the expected 1.44.
Does the
y
-intercept appear to be correct?
Yes, the intercept is close to -6, compared to the expected -6.04.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
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General Contents
Tangent Line Contents
Implicit Differentiation Contents
Index