Math Problem Statement
Derivative of y=xcoth(1+x^2)
Solution
To find the derivative of the function , we will apply the product rule and the chain rule.
Given:
Step 1: Apply the Product Rule
The product rule states: where and .
Step 2: Differentiate Each Component
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Differentiate :
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Differentiate : To differentiate , we use the chain rule. The derivative of is , so: The derivative of with respect to is: Therefore:
Step 3: Combine Using the Product Rule
Now, substitute back into the product rule: Simplifying, we get:
So, the derivative is:
Would you like more details or have any questions?
Here are five related questions to consider:
- What is the derivative of with respect to ?
- How would the derivative change if the function was instead?
- Can you find the second derivative of the function ?
- What are the properties of the hyperbolic cotangent function ?
- How would you differentiate without the factor?
Tip: Understanding the chain rule is crucial when dealing with composite functions, especially with hyperbolic functions.
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Math Problem Analysis
Mathematical Concepts
Differentiation
Hyperbolic Functions
Chain Rule
Product Rule
Formulas
Product rule
Chain rule
Theorems
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Suitable Grade Level
Undergraduate