Understanding the Derivative of Arcsin(x)

The video tutorial provides a detailed proof of the derivative of arcsin(x) with respect to x, showing that it equals 1 / sqrt(1 - x^2). The proof begins by modeling the problem using a right triangle, identifying the relationships between the sides and angles. Implicit differentiation is then applied to derive dy/dx, using the chain rule. The proof is verified by examining the graph of arcsin(x), confirming that the derivative is positive wherever it is defined, indicating an increasing function.

What is the derivative of arcsin(x) with respect to x?

• 1 / sqrt(1 - x^2)

• 1 / (1 + x^2)

• x / sqrt(1 - x^2)

• sqrt(1 - x^2)

In the context of the right triangle, what does the hypotenuse represent when modeling arcsin(x)?

• The square root of 1 - x^2

• The angle y

• The value x

• The number 1

Which trigonometric function is equal to the ratio of the adjacent side to the hypotenuse in a right triangle?

• Tangent

• Cosine

• Secant

• Sine

In which quadrants is the cosine function positive?

• First and fourth

• Second and third

• Third and fourth

• First and second

What is the result of differentiating sin(y) with respect to x using implicit differentiation?

• cos(y) * dy/dx

• sin(y) * dy/dx

• 1 / cos(y)

• dy/dx

How can 1 / cos(y) be expressed using a reciprocal identity?

• Cosecant

• Secant

• Sine

• Tangent

Where is the derivative of arcsin(x) not defined?

• At x = 0.5

• At x = 0

• At x = -0.5

• At x = 1 and x = -1

What does a positive derivative indicate about the function's behavior?

• The function is constant

• The function is decreasing

• The function is increasing

• The function is undefined

What is the significance of the slope of the tangent line being positive?

• The function is constant

• The function is increasing

• The function is decreasing

• The function is undefined

What does the graph of f(x) = arcsin(x) illustrate about the derivative?

• The derivative is zero everywhere

• The derivative is undefined everywhere

• The derivative is positive except at x = 1 and x = -1

• The derivative is negative everywhere