Math Problem Statement
Solution
The given function is:
We need to find the derivative of with respect to . Let's differentiate step by step.
Step 1: Differentiate
The derivative of with respect to is:
Step 2: Use the Chain Rule
Here, . By the chain rule:
First, let's find :
Now substitute into the chain rule:
Step 3: Simplify the Expression
Simplify the term inside the square root:
Thus, the derivative becomes:
Final Answer:
The derivative of with respect to is:
Would you like further details or have any questions?
Here are 5 related questions to further explore this topic:
- How would you find the second derivative of this function?
- What are the conditions for the domain of ?
- How does this derivative behave as approaches certain critical values (e.g., )?
- Can we derive this result using implicit differentiation?
- How would the derivative change if we had for some constant ?
Tip:
When differentiating inverse trigonometric functions, it's important to remember to apply the chain rule whenever the argument is not a simple variable.
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Math Problem Analysis
Mathematical Concepts
Differentiation
Inverse Trigonometric Functions
Chain Rule
Formulas
d/dx [arcsin(u)] = 1 / √(1 - u^2)
Chain Rule: dy/dx = dy/du * du/dx
Theorems
Derivative of Inverse Sine Function
Chain Rule in Differentiation
Suitable Grade Level
Grades 11-12 or University Calculus