How to Find the Second Derivative of Inverse Sine

TL;DR

To find the second derivative of the inverse sine function, start by determining the first derivative as 1 over the square root of 1 minus x squared. Apply the power rule to rewrite this as a negative exponent and differentiate, resulting in the second derivative expressed as x over (1 minus x squared) raised to the power of 3/2.

Transcript

okay we are going to find the second derivative of y is equal to the inverse sine function x where it's also called the arc sine x and for the second derivative that means i just have to take the derivative of this twice for the first derivative derivative arc sine x this is something that you should remember from the table you're going to derive i... Read More

Key Insights

• 🗂️ The first derivative of the inverse sine function can be derived from the formula 1 divided by the square root of 1 minus x squared.
• ✊ The second derivative of the inverse sine function can be found using the power rule by rewriting the equation as a power with a negative exponent.
• 🍉 Simplifying the second derivative equation involves multiplying the derivative of the inside function by -2x and rearranging the terms.

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.