(Q13.) So, you think you can take the derivative?  Summary and Q&A
TL;DR
The video explains how to find the second derivative of the inverse sine function using a simplified method.
Questions & Answers
Q: How do you find the first derivative of the inverse sine function?
The first derivative of the inverse sine function is found using the formula 1 divided by the square root of 1 minus x squared. This formula can be derived from the table or memorized.
Q: What is the advantage of rewriting the second derivative equation as a power with a negative exponent?
Rewriting the equation as a power with a negative exponent allows us to use the power rule to find the second derivative. It simplifies the calculation process and eliminates the need for the quotient rule.
Q: How do you simplify the second derivative equation further?
To simplify the second derivative equation, we multiply the derivative of the inside function (1 minus x squared) by the negative 2x. The negative sign and the 2x cancel out. The x term goes to the numerator, and the negative exponents are brought down to the denominator.
Q: What is the final equation for the second derivative of the inverse sine function?
The final equation for the second derivative is x divided by [(1 minus x squared) raised to the power of 3/2]. The numerator remains as x, while the denominator becomes the square root of (1 minus x squared) cubed.
Summary & Key Takeaways

The video demonstrates finding the first derivative of the inverse sine function, which is 1 divided by the square root of 1 minus x squared.

The second derivative of the inverse sine function is calculated using the power rule, resulting in an equation with a positive exponent.

Simplifying the equation further, the second derivative is written as x over the square root of 1 minus x squared raised to the power of 3/2.