Question

For the following exercises, find the derivatives tor the functions.\(\cosh ^{-1}\left(x^{3}\right)\)

Short answer

The derivative is \( \frac{3x^2}{\sqrt{x^6 - 1}} \).

Step-by-step solution

Write Down the Function and Identify the Outer Function

The function given is \( f(x) = \cosh^{-1}(x^3) \). Here, the outer function is \( \cosh^{-1}(u) \) where \( u = x^3 \). We will need to use the chain rule to find its derivative.

Apply the Chain Rule

The chain rule states that \( \frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} \). We need to first find \( \frac{df}{du} \) and \( \frac{du}{dx} \).

Differentiate the Outer Function with Respect to u

To find \( \frac{d}{du}[\cosh^{-1}(u)] \), use the derivative formula for the inverse hyperbolic cosine. The derivative is \( \frac{1}{\sqrt{u^2 - 1}} \). Thus, \( \frac{df}{du} = \frac{1}{\sqrt{(x^3)^2 - 1}} = \frac{1}{\sqrt{x^6 - 1}} \).

Differentiate the Inner Function with Respect to x

The inner function is \( u = x^3 \). So, the derivative \( \frac{du}{dx} = \frac{d}{dx}[x^3] = 3x^2 \).

Combine Using the Chain Rule

Substitute \( \frac{df}{du} \) and \( \frac{du}{dx} \) into the chain rule formula: \( \frac{df}{dx} = \frac{1}{\sqrt{x^6 - 1}} \cdot 3x^2 = \frac{3x^2}{\sqrt{x^6 - 1}} \).

Write the Final Answer

The derivative of the function \( \cosh^{-1}(x^3) \) is \( \frac{3x^2}{\sqrt{x^6 - 1}} \).

Key concepts

Inverse Hyperbolic Functions

Inverse hyperbolic functions are similar to trigonometric functions, but they involve hyperbolic functions such as hyperbolic sine, cosine, and tangent. The function

• Inverse hyperbolic cosine is denoted as \( \cosh^{-1}(x) \).
• It is defined for values \( x \geq 1 \) and considered an essential part of calculus and real analysis.

Understanding these functions is crucial as they appear in various areas such as engineering, physics, and mathematics.
They are often used to model behavior just like their trigonometric counterparts, but in hyperbolic settings instead.
When you differentiate inverse hyperbolic functions, specific formulas are employed for accuracy.
The formula for the derivative of \( \cosh^{-1}(x) \) is given by:\[\frac{d}{dx}[ \cosh^{-1}(x) ] = \frac{1}{\sqrt{x^2 - 1}}.\]
This formula is used when calculating how these functions change with an alteration in their input values.

Chain Rule

The chain rule is a powerful tool in calculus used to find the derivative of composite functions. A composite function essentially involves one function nested inside another.
In our problem,

• The function \( f(x) = \cosh^{-1}(x^3) \) is composed of the outer function \( \cosh^{-1}(u) \) and the inner function \( u = x^3 \).

The chain rule states that:\[\frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx}.\]
This tells us to take the derivative of the outer function first, then multiply it by the derivative of the inner function.
For beginners, it's helpful to think of it as peeling off layers until you reach the core, solving them, and then reassembling.

Differentiation Techniques

Differentiation techniques are methods used to find the derivative of different kinds of functions. Simple functions like polynomials have straightforward derivatives, whereas more complex functions, like our problem involving inverse hyperbolic functions, require several steps.
Here are the steps we followed in the original example:

• Identify the composite functions involved.
• Differentiation of inner and outer functions: We started by differentiating the inverse hyperbolic cosine, followed by differentiating the inner function \( x^3 \).
• Apply the Chain Rule: These derivatives were combined to get the complete derivative.

In our case, the derivative was computed in stages and then compiled using the chain rule.
Employing these methods gives precision and efficiency in differentiating a wide range of mathematical expressions.