Math Problem Statement
Solution
We are given the function , and the task is to find the derivative of .
Step-by-Step Solution:
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The derivative of the constant is , so we can focus on differentiating .
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The derivative of with respect to is given by: where in this case.
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Now, calculate the derivative of :
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Applying the chain rule, we get:
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Simplify the expression inside the square root: So, the derivative becomes:
Thus, the derivative of is:
Let me know if you need any further details!
Here are five related questions for you to consider:
- How does the derivative of the function change if the argument involves different powers of ?
- Can you compute the second derivative of the function ?
- What is the derivative of evaluated at ?
- How would you find the critical points of the function ?
- How does the behavior of the function change as ?
Tip: When differentiating composite functions like , always remember to use the chain rule effectively!
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Chain Rule
Inverse Trigonometric Functions
Formulas
d/dx [arcsin(u)] = 1 / sqrt(1 - u^2) * du/dx
d/dx [e^(4x)] = 4e^(4x)
Theorems
Chain Rule
Derivative of Inverse Trigonometric Functions
Suitable Grade Level
University/Advanced High School (Grades 11-12)