Question

For the following exercises, find the derivatives tor the functions.\(\tanh ^{-1}(4 x)\)

Short answer

The derivative is \(\frac{4}{1-16x^2}\).

Step-by-step solution

Identify Function

The given function is \(f(x) = \tanh^{-1}(4x)\). To find the derivative, we'll apply the chain rule.

Recall Derivative of Inverse Hyperbolic Tangent

The derivative of \(\tanh^{-1}(u)\) with respect to \(u\) is \(\frac{1}{1-u^2}\). We'll apply this to \(u = 4x\), but first, verify if \(|4x| < 1\) for the domain of the function, which is necessary to ensure the derivative exists.

Apply the Chain Rule

The chain rule states that \(\frac{d}{dx}[g(h(x))] = g'(h(x)) \cdot h'(x)\). Here, \(g(u) = \tanh^{-1}(u)\) and \(h(x) = 4x\). So, \(g'(u) = \frac{1}{1-u^2}\) and \(h'(x) = 4\).

Compute the Derivative

Using the chain rule, the derivative is \(f'(x) = \frac{1}{1-(4x)^2} \cdot 4\). Simplify the expression to obtain \(f'(x) = \frac{4}{1-16x^2}\).

Simplify the Result

The final expression for the derivative is \(f'(x) = \frac{4}{1-16x^2}\). This result provides the slope of the tangent line to the curve at any point within its domain.

Key concepts

Inverse Hyperbolic Functions

Inverse hyperbolic functions are counterparts to inverse trigonometric functions, specifically designed to handle hyperbolic functions like sinh, cosh, and tanh. The focus here is on the inverse hyperbolic tangent, denoted as \( \tanh^{-1}(x) \). This function is useful in mathematics due to its relationships in various branches of calculus and differential equations.

The inverse hyperbolic tangent function, \( \tanh^{-1}(x) \), relates to the regular hyperbolic tangent function \( \tanh(x) \), satisfying the equation \( y = \tanh^{-1}(x) \) or equivalently \( x = \tanh(y) \). The primary domain of \( \tanh^{-1}(x) \) is \(-1 < x < 1\), ensuring that the expression is valid and the function behaves appropriately throughout its intended range.

This implies that any problem involving the derivative of \( \tanh^{-1}(4x) \) should first confirm whether \(|4x| < 1\) to ensure the expression's validity. This check is vital because the function becomes undefined for \(|x| \geq 0.25\) due to the values of \(4x\), helping ensure the mathematical integrity throughout the problem-solving process.

Chain Rule

The chain rule is a fundamental principle in calculus, especially when differentiating composite functions. Essentially, it's a technique employed to differentiate functions nested within each other, following the formula: \( \frac{d}{dx}[g(h(x))] = g'(h(x)) \cdot h'(x) \).

In the given problem, the function \(f(x) = \tanh^{-1}(4x)\) is a prime example of where the chain rule shines. You have the outer function \(g(u) = \tanh^{-1}(u)\) and the inner function \(h(x) = 4x\).

To apply the chain rule in this case:

• First, compute the derivative of the outer function \(g'(u)\)—which is \(\frac{1}{1-u^2}\) when \( g(u) = \tanh^{-1}(u) \).


• Next, find the derivative of the inner function \(h(x)\), yielding \(h'(x) = 4\).


• Finally, multiply these derivatives together: \(f'(x) = g'(h(x)) \cdot h'(x)\), resulting in \( \frac{1}{1-(4x)^2} \cdot 4 \).

This straightforward application of the chain rule allows for a smooth calculation of otherwise complex derivative problems, showcasing its versatility in analyzing diverse mathematical scenarios.

Calculus Problem Solving

Calculus problem solving often involves strategically breaking down complex problems into manageable steps. By using derivatives as a tool, we are able to comprehend and model the behavior of functions, shedding light on their dynamics.

In solving for the derivative of \( \tanh^{-1}(4x) \), the process begins with understanding each component involved: the inverse hyperbolic function and the material properties of derivatives themselves. Applying the chain rule to take into account composite structures is crucial, as it simplifies the derivative calculation into manageable steps.

To solve calculus problems efficiently:

• Identify the function and its composite elements.


• Recall necessary derivative rules, such as those for inverse hyperbolic functions.


• Apply the appropriate calculus principles like the chain rule, working through the mathematics step-by-step.


• Verify the final result to ensure it aligns with domain constraints and provides meaningful insights.

Successful calculus problem solving is not just about arriving at the correct numerical result. It is equally about understanding the underlying mathematical principles and being able to apply them effectively across diverse problem scenarios.