91Ó°ÊÓ

In calculus, the product rule is a fundamental principle used to differentiate functions that are the product of two or more functions. The rule states that if you have a function \( y = uv \), where \( u \) and \( v \) are both differentiable functions of \( t \), then the derivative of \( y \) with respect to \( t \) is given by

• \( (uv)' = u'v + uv' \)

In the given exercise, the function \( y = (1-t^2) \coth^{-1} t \) is composed of two parts: \( u = 1-t^2 \) and \( v = \coth^{-1} t \). To find the derivative, we need to apply the product rule.

First, differentiate each part separately:

• The derivative of \( u \), \( u' = -2t \), since \( u = 1-t^2 \).
• The derivative of \( v \), using known derivatives of inverse hyperbolic functions, is \( v' = -\frac{1}{1-t^2} \).

Then plug these into the formula:

• \( y' = (-2t) (\coth^{-1} t) + (1-t^2) \left(-\frac{1}{1-t^2}\right) \)

This simplifies to become the final derivative expression. Using the product rule is key in breaking down and managing more complex expressions.

Inverse hyperbolic functions are similar to inverse trigonometric functions but relate to hyperbolic functions. For instance, \( \coth^{-1} t \) is the inverse hyperbolic cotangent. It finds use in various fields of math, including calculus and integration problems.

In the context of differentiation, knowing the derivatives of these functions is crucial:

• The derivative of \( \coth^{-1} t \) is \( -\frac{1}{1-t^2} \).

This derivative comes in handy when dealing with problems like our exercise, where we have to differentiate an expression involving \( \coth^{-1} t \). Understanding these derivatives allows for seamless application of rules like the product rule, making it easier to tackle derivative problems efficiently.

A good practice is to familiarize oneself with derivatives of other inverse hyperbolic functions such as \( \sinh^{-1} t \), \( \cosh^{-1} t \) , and so on, as they are frequently encountered in calculus.

Differentiation is the process of finding the derivative of a function, which gives us the rate of change of a function with respect to a variable. Differentiation techniques include several methods such as the product rule, quotient rule, chain rule, and implicit differentiation.

Each technique serves different types of functions. The choice of which technique to use depends largely on the form of the function you are trying to differentiate:

• Product Rule: As discussed, used for products of functions.
• Quotient Rule: Used when functions are divided by each other.
• Chain Rule: Applies when functions are nested within one another.
• Implicit Differentiation: Applies to equations not solved for one of the variables.

In exercises like the one provided, mastery of these rules and knowing when to apply them can greatly simplify the problem-solving process.

Practice these techniques regularly to become more comfortable with differentiating any function you encounter. Differentiating is a building-block concept that will form the foundation for higher-level math topics such as integration, solving differential equations, and optimization problems.