Derivatives of Inverse Trigonometric Functions

This video tutorial focuses on differentiating inverse trigonometric functions. It covers the derivatives of arc sine, arc cosine, arc tangent, and arc secant functions. The instructor provides formulas and step-by-step examples for each function, including arc sine of x cubed, arc cosine of 5x - 9, arc tangent of square root x, and arc secant of x to the fourth power. The tutorial emphasizes understanding the formulas and applying them to find derivatives efficiently.

What is the derivative of arc sine of x cubed?

• x cubed divided by the square root of 1 minus x squared

• 3x squared divided by the square root of 1 minus x to the sixth power

• x squared divided by the square root of 1 minus x cubed

• 3x cubed divided by the square root of 1 minus x to the fourth power

Which formula is used to find the derivative of arc cosine of a function?

• u prime divided by 1 plus u squared

• u prime divided by the absolute value of u times the square root of u squared minus one

• u prime divided by the square root of 1 minus u squared

• Negative u prime divided by the square root of 1 minus u squared

What is the derivative of arc cosine of 5x - 9?

• Negative 5 divided by the square root of 1 minus (5x - 9) squared

• 5 divided by 1 plus (5x - 9) squared

• Negative 5 divided by 1 plus (5x - 9) squared

• 5 divided by the square root of 1 minus (5x - 9) squared

What is the derivative of arc tangent of square root x?

• 1 over 2x divided by 1 plus x

• 1 over 2 square root x divided by 1 plus x

• Square root x divided by 1 plus x

• Square root x divided by 2x times 1 plus x

Which formula is used to find the derivative of arc tangent of a function?

• u prime divided by the absolute value of u times the square root of u squared minus one

• u prime divided by 1 plus u squared

• u prime divided by the square root of 1 minus u squared

• Negative u prime divided by the square root of 1 minus u squared

What is the simplified form of the derivative of arc tangent of square root x?

• Square root x divided by 1 plus x

• 1 over 2 square root x divided by 1 plus x

• 1 over 2x divided by 1 plus x

• Square root x divided by 2x times 1 plus x

What is the derivative of arc secant x to the fourth power?

• 4x cubed divided by the square root of x to the eighth minus one

• 4 divided by the square root of x to the eighth minus one

• 4 divided by x times the square root of x to the eighth minus one

• 4x cubed divided by the absolute value of x to the fourth times the square root of x to the eighth minus one

Which formula is used to find the derivative of arc secant of a function?

• u prime divided by 1 plus u squared

• Negative u prime divided by the square root of 1 minus u squared

• u prime divided by the square root of 1 minus u squared

• u prime divided by the absolute value of u times the square root of u squared minus one

How do you simplify the expression for the derivative of arc secant x to the fourth power?

• Add x to both the numerator and denominator

• Multiply the top and bottom by x squared

• Cancel three of the four x variables, leaving one in the denominator

• Subtract x from both the numerator and denominator

What is the role of the absolute value in the derivative of arc secant?

• It ensures the result is always positive

• It is necessary for all values of x

• It can be omitted if the term is always positive

• It is used to simplify the expression