Math Problem Statement
Solution
To find the second derivative of the function given:
we'll go through the following steps:
Step 1: Find (the first derivative)
We use the quotient rule for differentiation, which states that if , then
For our function:
- and
- and
Thus,
So, the first derivative is:
Step 2: Find (the second derivative)
Now, we differentiate again. We can rewrite it as:
Using the chain rule:
Final Answer
The second derivative of the function is:
Would you like more details or have any questions?
Here are some related questions to consider:
- How does the quotient rule work for finding derivatives?
- Why do we use the chain rule when differentiating ?
- How would the derivatives change if had a different denominator?
- What is the significance of the sign of in terms of concavity?
- Can have any inflection points?
Tip: When differentiating functions with complex denominators, rewriting them in exponent form can simplify the process, especially for higher derivatives.
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Quotient Rule
Chain Rule
Formulas
Quotient Rule: (g(x)/h(x))' = (g'(x)h(x) - g(x)h'(x)) / [h(x)]^2
Chain Rule: d/dx [f(g(x))] = f'(g(x)) * g'(x)
Theorems
Quotient Rule for Differentiation
Chain Rule for Differentiation
Suitable Grade Level
Grades 11-12
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