Math Problem Statement
Find the derivative of the function. f(x) = sin^−1(4x)
Solution
To find the derivative of the function , we will apply the chain rule along with the known derivative formula for the inverse sine function.
Step 1: Apply the derivative formula for
The derivative of with respect to is:
Step 2: Use the chain rule
Now, since the function is , we recognize that is the inside function. We use the chain rule to differentiate.
So, applying the chain rule:
Step 3: Differentiate
The derivative of with respect to is simply 4.
Thus, the derivative becomes:
Final Answer:
Would you like more details on the derivative process, or do you have any questions about this solution?
Here are some related questions to explore:
- How do you differentiate inverse trigonometric functions with other compositions (e.g., )?
- What is the domain of the function ?
- How do you apply the chain rule in more complex functions?
- Can you explain how to simplify the derivative of a function involving square roots?
- What is the physical significance of the derivative of inverse sine in a real-world scenario?
Tip: Always check the domain of the function before differentiating, as it can impact the validity of the derivative for certain values of .
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Inverse Trigonometric Functions
Chain Rule
Formulas
Derivative of sin^−1(x) = 1 / sqrt(1 - x^2)
Chain Rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
Theorems
Chain Rule
Derivative of Inverse Trigonometric Functions
Suitable Grade Level
Grades 11-12