Derivatives of Hyperbolic Functions

The video tutorial explains how to determine the derivatives of hyperbolic functions using the chain rule. It provides two examples: the first involves finding the derivative of the hyperbolic tangent of 2x, and the second involves the derivative of the hyperbolic cosine raised to the fifth power. The tutorial emphasizes recognizing composite functions and applying the chain rule effectively, highlighting the importance of identifying inner and outer functions in the process.

What is the first step in finding the derivative of a composite function involving hyperbolic functions?

• Identify the outer function

• Use the quotient rule

• Apply the product rule

• Identify the inner function

In the function f(x) = hyperbolic tangent(2x), what is the inner function?

• hyperbolic tangent

• 2x

• hyperbolic secant

• x^2

What is the derivative of the outer function in f(x) = hyperbolic tangent(2x)?

• hyperbolic secant squared

• hyperbolic tangent

• hyperbolic sine

• hyperbolic cosine

What is the result of applying the chain rule to f(x) = hyperbolic tangent(2x)?

• hyperbolic sine(2x)

• 2 hyperbolic secant squared(2x)

• 2 hyperbolic tangent(2x)

• hyperbolic cosine(2x)

What is the derivative of f(x) = hyperbolic cosine^5(x) primarily concerned with?

• Product rule

• Quotient rule

• General power rule

• Sum rule

In the function f(x) = hyperbolic cosine^5(x), what is the outer function?

• hyperbolic sine

• power function

• x^5

• hyperbolic cosine

What is the derivative of the inner function in f(x) = hyperbolic cosine^5(x)?

• hyperbolic sine

• hyperbolic tangent

• hyperbolic secant

• hyperbolic cosine

How is the chain rule applied in the derivative of f(x) = hyperbolic cosine^5(x)?

• By differentiating the outer function only

• By differentiating the inner function only

• By differentiating both the inner and outer functions

• By using the product rule

What is the role of the chain rule in finding derivatives of composite functions?

• It simplifies the function

• It helps differentiate the inner and outer functions separately

• It eliminates the need for other rules

• It only applies to trigonometric functions

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

• hyperbolic tangent^5(x)

• 5 hyperbolic sine^4(x) hyperbolic cosine(x)

• hyperbolic secant^5(x)

• 5 hyperbolic cosine^4(x) hyperbolic sine(x)