(Q13.) So, you think you can take the derivative? | Summary and Q&A

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April 13, 2014
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blackpenredpen
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(Q13.) So, you think you can take the derivative?

TL;DR

The video explains how to find the second derivative of the inverse sine function using a simplified method.

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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.

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