Find the derivative of the function.
[tex]Y=7 \operatorname{SIN}^{-1}\left(X^2\right)[/tex]
Answer (1)
Apply the chain rule to find the derivative of Y = 7 SIN − 1 ( X 2 ) .
Recall that the derivative of SIN − 1 ( u ) is 1 − u 2 1 ⋅ d x d u .
Find d x d u where u = X 2 , which gives d x d u = 2 X .
The derivative is y ′ = 1 − x 4 14 x .
Explanation
Problem Analysis We are given the function Y = 7 SIN − 1 ( X 2 ) and asked to find its derivative. This problem involves finding the derivative of an inverse trigonometric function, specifically the inverse sine function. We will need to apply the chain rule.
Derivative of Inverse Sine Recall that the derivative of SIN − 1 ( u ) with respect to x is 1 − u 2 1 ⋅ d x d u . In our case, u = X 2 , so d x d u = 2 X .
Applying the Chain Rule Applying the chain rule, we have: d X d Y = 7 ⋅ 1 − ( X 2 ) 2 1 ⋅ ( 2 X ) = 1 − X 4 14 X Thus, the derivative of Y with respect to X is 1 − X 4 14 X .
Final Answer Comparing our result with the given options, we see that the correct answer is y ′ = 1 − x 4 14 x .
Examples
Imagine you are designing a navigation system and need to calculate the angle to a certain point. The inverse sine function helps you find that angle, and knowing its derivative allows you to understand how sensitive the angle is to changes in position, which is crucial for accurate navigation.
