Math Problem Statement

Find the derivative of the function. f(x) = sin^−1(4x)

Solution

To find the derivative of the function f(x)=sin1(4x)f(x) = \sin^{-1}(4x), we will apply the chain rule along with the known derivative formula for the inverse sine function.

Step 1: Apply the derivative formula for sin1(x)\sin^{-1}(x)

The derivative of sin1(x)\sin^{-1}(x) with respect to xx is: ddx[sin1(x)]=11x2\frac{d}{dx} \left[ \sin^{-1}(x) \right] = \frac{1}{\sqrt{1 - x^2}}

Step 2: Use the chain rule

Now, since the function is f(x)=sin1(4x)f(x) = \sin^{-1}(4x), we recognize that 4x4x is the inside function. We use the chain rule to differentiate.

So, applying the chain rule:

ddx[sin1(4x)]=11(4x)2ddx(4x)\frac{d}{dx} \left[ \sin^{-1}(4x) \right] = \frac{1}{\sqrt{1 - (4x)^2}} \cdot \frac{d}{dx}(4x)

Step 3: Differentiate 4x4x

The derivative of 4x4x with respect to xx is simply 4.

Thus, the derivative becomes: f(x)=4116x2f'(x) = \frac{4}{\sqrt{1 - 16x^2}}

Final Answer:

f(x)=4116x2f'(x) = \frac{4}{\sqrt{1 - 16x^2}}


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:

  1. How do you differentiate inverse trigonometric functions with other compositions (e.g., tan1(5x)\tan^{-1}(5x))?
  2. What is the domain of the function sin1(4x)\sin^{-1}(4x)?
  3. How do you apply the chain rule in more complex functions?
  4. Can you explain how to simplify the derivative of a function involving square roots?
  5. 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 xx.

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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