Derivatives and Inverse Functions

The video tutorial explains how to find the inverse of a function and its derivative without explicitly finding the inverse function. It uses the function f(x) = x^3 - 2/x and demonstrates how to determine f inverse of 7 by finding an x value that satisfies f(x) = 7. The tutorial also covers how to find the derivative of the inverse function at a specific point using the reciprocal of the derivative of the original function. Graphical representations are used to illustrate the concepts, and the tutorial concludes with a summary of the findings.

What is the function f(x) given in the problem?

• x squared minus two divided by x

• x cubed minus two divided by x

• x cubed plus two divided by x

• x squared plus two divided by x

What is the relationship between f(a) = b and f inverse?

• f inverse of b equals a

• f inverse of a equals b

• f inverse of a equals a

• f inverse of b equals b

If f(2) equals 7, what is f inverse of 7?

• Negative one

• 2

• 1

• 7

What is the formula for the derivative of f inverse at x = b?

• 1 over f inverse of b

• f prime of a

• 1 over f prime of a

• f inverse of b

What is the expression for f prime of x?

• 3x minus 2 divided by x

• 3x squared plus 2 divided by x squared

• 3x squared minus 2 divided by x squared

• 3x plus 2 divided by x

What is the value of f prime of 2?

• 12

• 25 halves

• 2

• 7

What is the slope of the tangent line to the inverse function at x = 7?

• 25 halves

• 7

• 12

• 2 twenty-fifths

What is the reciprocal of the slope of the tangent line to f(x) at the point (2, 7)?

• 7

• 1

• 2 twenty-fifths

• 25 halves

What is the derivative of f inverse at x = 7?

• 25 halves

• 2 twenty-fifths

• 1

• 7

What is the slope of the tangent line to the original function at x = 2?

• 2 twenty-fifths

• 1

• 7

• 25 halves