Question

For the following exercises, find the derivatives tor the functions.\(e^{\sinh ^{-1}(x)}\)

Short answer

The derivative is \( y'(x) = e^{\sinh^{-1}(x)} \cdot \frac{1}{\sqrt{x^2 + 1}} \).

Step-by-step solution

Define the Function

First, we clearly state the function that we are going to differentiate, which is given as: \[y(x) = e^{\sinh^{-1}(x)}\]

Differentiate Using the Chain Rule

To differentiate the function, we'll apply the chain rule. Recall that for a function of the form \( e^{u(x)} \), its derivative is \( e^{u(x)} \cdot u'(x) \). Here, \( u(x) = \sinh^{-1}(x) \).

Differentiate the Inverse Hyperbolic Sine

Differentiate \( u(x) \). The derivative of \( \sinh^{-1}(x) \) is \( \frac{1}{\sqrt{x^2 + 1}} \).

Apply the Chain Rule

Now, substitute back into the chain rule formula:\[y'(x) = e^{\sinh^{-1}(x)} \cdot \frac{1}{\sqrt{x^2 + 1}}\]This represents the derivative of the given function.

Key concepts

Derivatives

Derivatives are a fundamental concept in calculus, representing how a function changes as its input changes. In simple terms, a derivative gives us the slope of a function at any given point. This is incredibly useful to understand the behavior of functions, especially when trying to find rates of change or optimizing values.

• The notation for the derivative of a function \(y\) with respect to \(x\) is \(y'(x)\) or \(\frac{dy}{dx}\).


• A derivative can inform us about whether a function is increasing or decreasing, and at what rate.


• Knowing the derivative of a function helps in sketching graphs, studying motion, and analyzing real-world phenomena.

When you write a function in a specific form, such as an exponential, polynomial, or trigonometric function, it helps to know the formulas for derivatives of these forms. This allows you to efficiently find the derivative without re-deriving every formula.
Practicing finding derivatives helps students understand how different kinds of functions behave.

Chain Rule

The chain rule is a critical technique for finding the derivative of composite functions, that is, functions composed of one function operating on another. It allows us to differentiate complex functions where one function is nested inside another.
Consider a simple example, the function \(y = (3x+2)^5\), which can be seen as two functions: the outer function \(u(x) = u^5\) and the inner function \(u(x) = 3x+2\).

• The chain rule states that if you have a composite function \(y(x) = f(g(x))\), then the derivative \(y'(x)\) is given by \(f'(g(x)) \cdot g'(x)\).


• In this example, if you find \(g(x)\) and \(g'(x)\) and multiply it by the derivative of \(f(g(x))\), you essentially "chain" them together.

Applying the chain rule requires practice to identify inner and outer functions correctly. In more advanced problems, you might deal with functions like inverse hyperbolic functions or logarithmic ones, where using the chain rule can become very handy and essential.

Inverse Hyperbolic Functions

Inverse hyperbolic functions provide a way to assess specific ratios or angles associated with hyperbolic trigonometric functions. They have more complex derivatives compared to regular trigonometric functions.
Some of the common inverse hyperbolic functions are:\(\sinh^{-1}(x)\), \(\cosh^{-1}(x)\), and \(\tanh^{-1}(x)\).

• The derivative of the inverse hyperbolic sine, \(\sinh^{-1}(x)\), is \(\frac{1}{\sqrt{x^2 + 1}}\). This derivative is used in many calculus problems where inverse hyperbolic functions appear.


• These functions are particularly useful in solving integrals or working with logarithmic expressions related to hyperbolic functions.


• Being familiar with inverse hyperbolic functions and their derivatives is especially valuable since they often appear in more advanced calculus problems.

Understanding the properties and behaviors of inverse hyperbolic functions can provide deeper insights into the mechanics of calculus and enhance skills for tackling diverse mathematical challenges.