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Answer
Solution
It looks like you’re working through the differentiation of the function:
Let’s walk through the differentiation process step-by-step using the Product Rule. The Product Rule states that if you have a function
, then its derivative is:
Step 1: Identify
and
Let’s assign:
Step 2: Compute
and
Differentiate
and
with respect to
:
Step 3: Apply the Product Rule
Substitute
,
,
, and
into the Product Rule formula:
Plugging in the values:
Step 4: Simplify the Expression
Let’s simplify each term:
- First Term:
- Second Term:
Putting it all together:
Step 5: Combine the Terms (Optional)
If you want to combine the two terms into a single fraction:
Simplify the numerator:
So, the derivative becomes:
Final Answer
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Bonus Knowledge
Let’s kick off with a bit of historical context! The use of calculus to differentiate complex functions dates back to the work of Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. They introduced not only the concept of derivatives but also laid the groundwork for understanding the relationship between a function and its rate of change. Mathematical notation has evolved since then, but the core principles remain integral to studying functions like the one you’re working with!
Now, for some real-world application! The differentiation of functions like
doesn’t just stay in the math classroom; it has practical uses in various fields like physics, engineering, and economics. For instance, in real-world modeling situations such as optimizing production in factories or predicting the trajectory of projectiles, finding the maximum or minimum values through derivatives can lead to improved efficiency and better decision-making. Who knew calculus could help you hit your targets both literally and figuratively?