Trigonometric Functions and Derivatives

The video tutorial explores the derivative of the tangent function and its reciprocal, focusing on trigonometric identities and substitution methods. It verifies predictions about the behavior of the inverse tangent function, ensuring it is always positive and has no vertical asymptotes. The lesson concludes with a graphical representation of the gradient function, emphasizing its even nature and behavior at extremities.

What is the derivative of tan(x)?

• sin^2(x)

• tan^2(x)

• cos^2(x)

• sec^2(x)

What is the reciprocal of sec^2(x)?

• cos^2(x)

• tan^2(x)

• 1/cos^2(x)

• 1/sin^2(x)

Which identity helps in translating between sec and tan?

• tan^2(x) + sec^2(x) = 1

• 1 + tan^2(x) = sec^2(x)

• sin^2(x) + cos^2(x) = 1

• 1 + sec^2(x) = tan^2(x)

What is the derivative of tan inverse with respect to y?

• 1/(1 + x^2)

• 1/(1 + y^2)

• y/(1 + x^2)

• x/(1 + y^2)

Why does the function not have a vertical asymptote?

• Because x^2 can be zero

• Because x^2 is always positive

• Because x^2 can be negative

• Because x^2 is always zero

What is the smallest value that x^2 can take?

• 1

• 0

• 2

• -1

What happens to the function as x approaches infinity?

• It becomes negative

• It approaches one

• It approaches zero

• It becomes undefined

What is the nature of the function discussed in the final section?

• Both odd and even

• Neither odd nor even

• Even function

• Odd function

What is the gradient function of tan inverse?

• 1/(1 + x^2)

• 1/(1 - x^2)

• x/(1 - x^2)

• x/(1 + x^2)

What kind of curve is 1 + x^2?

• Hyperbola

• Parabola

• Linear

• Circle