Understanding Derivatives and Inverse Hyperbolic Functions

The video tutorial explains how to determine the derivative of a given function using product and chain rules. It simplifies the derivative expression and calculates the derivative at a specific point, x=3. The tutorial also demonstrates how to evaluate the derivative using both the T-84 and Desmos graphing calculators, providing a step-by-step guide to ensure accurate results.

What is the initial form of the function given in the problem?

• 3x inverse hyperbolic sine of x divided by 2 plus e square root of x squared minus 4

• 3x inverse hyperbolic cosine of x divided by 2 plus e square root of x squared minus 4

• 3x inverse hyperbolic sine of x divided by 2 minus e square root of x squared minus 4

• 3x inverse hyperbolic cosine of x divided by 2 minus e square root of x squared minus 4

What is the rewritten form of the function for differentiation?

• 3x inverse hyperbolic cosine of one-half x minus the quantity x squared minus 4 to the power of one-half

• 3x inverse hyperbolic sine of one-half x minus the quantity x squared minus 4 to the power of one-half

• 3x inverse hyperbolic cosine of one-half x plus the quantity x squared minus 4 to the power of one-half

• 3x inverse hyperbolic sine of one-half x plus the quantity x squared minus 4 to the power of one-half

Which rules are applied to find the derivative of the function?

• Product rule and quotient rule

• Chain rule and quotient rule

• Sum rule and chain rule

• Product rule and chain rule

What is the derivative of the first function, 3x, in the product rule?

• 2

• 0

• 1

• 3

In the differentiation process, what is the derivative of the inverse hyperbolic cosine function?

• 1 divided by the square root of u squared plus 1

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

• 1 divided by the square root of u squared plus 2

• 1 divided by the square root of u squared minus 2

What is the value of u prime when differentiating the inverse hyperbolic cosine function?

• 1

• 1/2

• 2

• 3

What is the simplified form of the derivative after combining fractions?

• 2x divided by the square root of x squared minus 4 plus 3 times inverse hyperbolic cosine of one-half x

• 2x divided by the square root of x squared minus 4 plus 3 times inverse hyperbolic sine of one-half x

• 3x divided by the square root of x squared minus 4 plus 2 times inverse hyperbolic sine of one-half x

• 3x divided by the square root of x squared minus 4 plus 2 times inverse hyperbolic cosine of one-half x

What is the final expression for f prime of x after simplification?

• 2x divided by the square root of x squared minus 4 plus 3 times inverse hyperbolic cosine of one-half x

• 2x divided by the square root of x squared minus 4 minus 3 times inverse hyperbolic cosine of one-half x

• 3x divided by the square root of x squared minus 4 minus x divided by the square root of x squared minus 4

• 3x divided by the square root of x squared minus 4 plus x divided by the square root of x squared minus 4

How is the derivative evaluated at x = 3 using a calculator?

• By entering 6 divided by the square root of 5 plus 3 times inverse hyperbolic cosine of 1.5

• By entering 6 divided by the square root of 5 plus 3 times inverse hyperbolic sine of 1.5

• By entering 6 divided by the square root of 5 minus 3 times inverse hyperbolic sine of 1.5

• By entering 6 divided by the square root of 5 minus 3 times inverse hyperbolic cosine of 1.5

What is the approximate value of f prime of 3?

• 6.5706

• 4.5706

• 5.5706

• 3.5706