Understanding Derivatives of Inverse Trigonometric Functions

The video tutorial explains how to find the derivative of the inverse tangent function. It begins with a review of the derivative of the tangent function and then applies the chain rule to find the derivative of the inverse tangent. The process involves expressing the derivative in terms of x using trigonometric identities. The final result is the derivative of the inverse tangent function expressed as 1/(1 + x^2).

What is the derivative of tangent of x with respect to x?

• Secant of x

• Sine of x

• Secant squared of x

• Cosine of x

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

• Set x equal to tangent of y

• Set x equal to inverse tangent of y

• Set y equal to inverse tangent of x

• Set y equal to tangent of x

Which rule is applied to differentiate the tangent of y with respect to x?

• Power Rule

• Quotient Rule

• Product Rule

• Chain Rule

What is the derivative of y with respect to x initially expressed as?

• Sine of y squared

• Cosine of y squared

• Secant of y squared

• Tangent of y squared

Why is it necessary to express the derivative in terms of x?

• To relate it to the original function

• To make it easier to integrate

• To avoid using trigonometric identities

• To simplify the expression

What identity is used to manipulate the expression involving cosine squared y?

• Pythagorean identity

• Reciprocal identity

• Double angle identity

• Sum of angles identity

What is the purpose of dividing by cosine squared y in the expression?

• To convert to a rational expression

• To eliminate sine

• To simplify the expression

• To introduce a tangent

What does the expression one over one plus tangent of y squared simplify to?

• One over one plus x squared

• One over x squared

• One plus x squared

• X squared

What is the final expression for the derivative of inverse tangent of x?

• One plus x squared

• One over one plus x squared

• X over one plus x squared

• One over x squared

What is the significance of the final result in the context of inverse trigonometric functions?

• It proves the derivative is zero

• It shows the derivative is always positive

• It relates the derivative to a simple algebraic expression

• It demonstrates the use of the product rule