Derivative of Inverse Hyperbolic Cosine Function

The video tutorial explains how to find the derivative of a composite function, specifically focusing on the function f(x) = 2 inverse hyperbolic cosine of 5x. It emphasizes the importance of recognizing the function as composite and applying the chain rule. The tutorial breaks down the process into identifying the inner function, applying the derivative formula for inverse hyperbolic cosine, and simplifying the result. The explanation highlights the use of the chain rule, where the derivative of the outer function is evaluated at the inner function, and the derivative of the inner function is calculated separately.

What is the main function we are trying to differentiate in this tutorial?

• f(x) = 2 inverse hyperbolic sine of 5x

• f(x) = 2 inverse hyperbolic cosine of 5x

• f(x) = 2 inverse hyperbolic tangent of 5x

• f(x) = 2 inverse hyperbolic secant of 5x

Why is it important to recognize that f(x) is a composite function?

• To apply the product rule

• To apply the quotient rule

• To apply the chain rule

• To apply the power rule

Which rule is emphasized in the explanation of the derivative formula?

• Chain rule

• Quotient rule

• Power rule

• Product rule

What is the inner function in the composite function f(x) = 2 inverse hyperbolic cosine of 5x?

• x/5

• 5x

• 2x

• x^2

What is the derivative of the inner function 5x with respect to x?

• x

• 5

• 1

• 0

What is the role of the factor U' in the derivative formula?

• It represents the derivative of the inner function

• It represents the original function

• It represents the derivative of the outer function

• It represents the integral of the function

What is the derivative of the inverse hyperbolic cosine function with respect to its argument?

• 1 / sqrt(u^2 - 1)

• 1 / sqrt(1 - u^2)

• 1 / (1 + u^2)

• 1 / (u^2 + 1)

How is the chain rule applied in the context of this function?

• By subtracting the derivative of the outer function from the derivative of the inner function

• By adding the derivative of the outer function to the derivative of the inner function

• By dividing the derivative of the outer function by the derivative of the inner function

• By multiplying the derivative of the outer function by the derivative of the inner function

What is the final expression for the derivative of f(x) = 2 inverse hyperbolic cosine of 5x?

• 5 / sqrt(25x^2 - 1)

• 1 / sqrt(25x^2 - 1)

• 2 / sqrt(25x^2 - 1)

• 10 / sqrt(25x^2 - 1)

What is the purpose of simplifying the derivative expression?

• To make it easier to integrate

• To make it easier to interpret and use

• To make it easier to differentiate again

• To make it easier to graph