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Differentiate implicitly to find

Ask by Riley Bowen. in the United States
Nov 25,2024

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The second derivative is given by the expression:

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To differentiate the equation implicitly, we start by differentiating both sides with respect to :
  1. The derivative of is .
  2. The derivative of requires the product rule: .
  3. The derivative of is .
  4. 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.

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