Math Problem Statement
Can you use as much detail as possible for how to solve
Solution
Let’s solve the problem step by step with detailed explanations.
Problem Statement
We are asked to find the derivative where:
Step 1: Understand the given expression
The function consists of two terms:
- The first term is , which is a linear function of .
- The second term is , which is a rational function.
To find , we need to differentiate both terms with respect to .
Step 2: Differentiate
The derivative of with respect to is straightforward:
Step 3: Differentiate
For the second term, we use the power rule. Rewrite as:
The power rule states that the derivative of is . Applying this rule to :
Step 4: Combine the results
Now, sum the derivatives of the two terms: Substitute the results:
Simplify:
Step 5: Verify the answer
From the choices given, the correct answer is:
Recap
- The derivative of is .
- The derivative of is .
- Adding these results gives .
Would you like further clarification or more examples?
5 Related Questions:
- What is the derivative of ?
- How do you find the second derivative of ?
- Can you explain the power rule in differentiation?
- What is the derivative of a general rational function ?
- How would this problem change if the function was ?
Tip:
Whenever dealing with terms like , rewrite them as to make differentiation easier!
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Math Problem Analysis
Mathematical Concepts
Differentiation
Power Rule
Rational Functions
Formulas
\(\frac{d}{dt}(t) = 1\)
\(\frac{d}{dt}(t^n) = n \cdot t^{n-1}\)
\(\frac{d}{dt}\left(\frac{9}{t}\right) = -\frac{9}{t^2}\)
Theorems
Power Rule of Differentiation
Suitable Grade Level
Grades 11-12 or Introductory Calculus