Understanding Derivatives of Inverse Trigonometric Functions

The video tutorial explains how to derive the derivative of the inverse cosine function. It begins with a recap of the inverse sine derivative and sets up the problem by defining y as the inverse cosine of x. The tutorial then derives the expression for dy/dx in terms of y and converts it to x using trigonometric identities. The final derivative is shown to be negative 1 over the square root of 1 minus x squared. The video concludes by comparing the derivatives of inverse sine and cosine, highlighting the only difference being the sign.

What is the derivative of the inverse sine of x?

• 1 over the square root of 1 minus x squared

• Negative 1 over the square root of 1 minus x squared

• 1 over 1 plus x squared

• Negative 1 over 1 plus x squared

What is the first step in finding the derivative of the inverse cosine of x?

• Use the Pythagorean identity

• Differentiate x with respect to y

• Set y equal to the inverse cosine of x

• Set y equal to the inverse sine of x

When differentiating x = cos(y), what is the derivative of cos(y) with respect to y?

• sin(y)

• Negative cos(y)

• Negative sin(y)

• cos(y)

How do you express dy/dx in terms of x after finding it in terms of y?

• Use the identity sin(y) = 1 - cos^2(y)

• Use the identity tan^2(y) + 1 = sec^2(y)

• Use the identity 1 + cot^2(y) = csc^2(y)

• Use the identity sin^2(y) + cos^2(y) = 1

What is the derivative of the inverse cosine of x?

• 1 over the square root of 1 minus x squared

• Negative 1 over the square root of 1 minus x squared

• 1 over 1 plus x squared

• Negative 1 over 1 plus x squared

Which trigonometric identity is used to convert dy/dx from terms of y to terms of x?

• sin^2(y) + cos^2(y) = 1

• tan^2(y) + 1 = sec^2(y)

• sin(y) = 1 - cos^2(y)

• 1 + cot^2(y) = csc^2(y)

What is the expression for sine of y in terms of cosine of y?

• Square root of 1 minus cosine squared of y

• Square root of 1 plus cosine squared of y

• 1 plus cosine squared of y

• 1 minus cosine squared of y

What is the main difference between the derivatives of inverse sine and inverse cosine?

• The derivative of inverse cosine is positive

• The derivative of inverse sine is negative

• The derivative of inverse sine is positive, and inverse cosine is negative

• Both derivatives are negative

What is the derivative of the inverse sine of x in terms of x?

• Negative 1 over the square root of 1 minus x squared

• 1 over 1 plus x squared

• 1 over the square root of 1 minus x squared

• Negative 1 over 1 plus x squared

What is the significance of the negative sign in the derivative of the inverse cosine?

• It shows the function is increasing

• It is a result of the Pythagorean identity

• It indicates a decrease in the function

• It is due to the derivative of sine being negative