Derivatives of Hyperbolic Functions

The video tutorial explains how to determine the derivatives of composite functions using the chain rule. It provides two examples: one involving the hyperbolic tangent function and another with the hyperbolic cosine function. The tutorial breaks down each step, showing how to identify the inner function and apply the chain rule to find the derivative. The first example focuses on f(x) = 4 tanh(3x), while the second example deals with f(x) = (cosh(6x))^4. The video concludes with a summary of the methods used.

What is the derivative of a composite function involving hyperbolic tangent?

• Square of hyperbolic secant times the derivative of the inner function

• Hyperbolic sine times the derivative of the inner function

• Hyperbolic cosine times the derivative of the inner function

• Square of hyperbolic sine times the derivative of the inner function

In the first example, what is the value of u prime when u equals 3x?

• 3

• 9

• 1

• 6

How do you express the derivative of the function 4 hyperbolic tangent 3x?

• 4 times the square of hyperbolic secant of 3x

• 12 times the square of hyperbolic secant of 6x

• 4 times the hyperbolic cosine of 3x

• 12 times the hyperbolic sine of 6x

What is the inner function in the second example involving hyperbolic cosine?

• Hyperbolic secant 6x

• Hyperbolic tangent 6x

• Hyperbolic cosine 6x

• Hyperbolic sine 6x

What is the derivative of hyperbolic cosine u with respect to x?

• Hyperbolic cosine u times u prime

• Hyperbolic sine u times u prime

• Hyperbolic tangent u times u prime

• Hyperbolic secant u times u prime

How is the derivative of the function u to the fourth expressed?

• 4 times u to the sixth times u prime

• 4 times u to the fifth times u prime

• 4 times u squared times u prime

• 4 times u cubed times u prime

What is the value of u prime when u equals hyperbolic cosine of 6x?

• 6 hyperbolic sine 6x

• 6 hyperbolic secant 6x

• 6 hyperbolic cosine 6x

• 6 hyperbolic tangent 6x

What is the simplified form of the derivative for the second example?

• 24 times the square of hyperbolic cosine 6x times hyperbolic sine 6x

• 12 times the cube of hyperbolic cosine 6x times hyperbolic sine 6x

• 24 times the cube of hyperbolic cosine 6x times hyperbolic sine 6x

• 12 times the square of hyperbolic cosine 6x times hyperbolic sine 6x

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

• It simplifies the process of finding derivatives of composite functions

• It helps in differentiating functions with multiple variables

• It is used only for trigonometric functions

• It is not applicable to hyperbolic functions