Question
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Tutor-Verified Answer
Answer
The second derivative
is given by the expression:
Solution
To find the second derivative
of the equation
, we will first differentiate the equation implicitly with respect to
.
Starting with the equation:
we differentiate both sides with respect to
:
Using the chain rule and product rule, we differentiate each term:
-
For
: -
For
: -
For
:
Putting it all together, we have:
Rearranging this gives:
Combining the terms involving
:
Now, isolating
:
Next, we differentiate
again to find
. We will use the quotient rule:
Let
and
. Then:
Using the quotient rule:
Now we need to find
and
:
-
For
: -
For
:
Substituting these into the expression for
:
Now substituting
into the equation will give us
in terms of
and
. However, this expression can be quite complex, so we will leave it in this form for clarity.
Thus, the second derivative is:
This is the implicit differentiation result for
.
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Bonus Knowledge
To differentiate the equation
implicitly, we start by differentiating both sides with respect to
:
- The derivative of
is . - The derivative of
requires the product rule: . - The derivative of
is . - The derivative of the constant
is .
Putting it all together gives us:
Now, combine like terms:
This allows us to solve for
:
Next, we need to differentiate
implicitly again to find
. We apply the quotient rule, which states that
's derivative is given by
. In our case, let
and
.
Differentiating
and
:
- For
:
- For
:
Putting it into the quotient rule gives:
After some simplification, you’ll arrive at the expression for
. Remember to substitute back
with its previous result if necessary!
Now, for a fun fact: This type of differentiation, while it may seem tedious, unlocks a world of physics! It is essential in determining acceleration in relation to velocity and can even help in fields such as engineering and economics to find rates of change in systems!
Additionally, understanding implicit differentiation can take you on a historical journey! It’s fascinating how calculus was developed during the 17th century by brilliant minds like Newton and Leibniz, who independently formulated much of what we use today! It revolutionized mathematics and paved the way for advancements in science and technology.