Question

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

Short answer

The derivative is \(-\csc(x)\).

Step-by-step solution

Identify the Composition of Functions

The function given is \(\tanh^{-1}(\cos(x))\). It is a composition of the inverse hyperbolic tangent function \(\tanh^{-1}(u)\) and the cosine function \(u = \cos(x)\). To differentiate it, we'll need to apply the chain rule.

Derivative of the Outer Function

Differentiate the outer function \(\tanh^{-1}(u)\) with respect to \(u\). The derivative is \(\frac{1}{1-u^2}\). Thus, the derivative of \(\tanh^{-1}(\cos(x))\) with respect to \(\cos(x)\) is \(\frac{1}{1-(\cos(x))^2}\).

Derivative of the Inner Function

Differentiate the inner function \(u = \cos(x)\) with respect to \(x\). The derivative of \(\cos(x)\) is \(-\sin(x)\).

Apply the Chain Rule

By the chain rule, the derivative of the composite function \(\tanh^{-1}(\cos(x))\) with respect to \(x\) is the product of the derivatives from Steps 2 and 3: \(-\frac{\sin(x)}{1-(\cos(x))^2}\).

Simplify the Expression

Since \(1-(\cos(x))^2 = \sin^2(x)\), the derivative can be simplified to \(-\frac{\sin(x)}{\sin^2(x)} = -\frac{1}{\sin(x)} = -\csc(x)\).

Key concepts

Chain Rule

The Chain Rule is an essential tool in calculus for differentiating composite functions. When you have a function nested inside another function, like \( f(g(x)) \), the chain rule comes into play.Start by differentiating the outer function while keeping the inner function unchanged. Next, multiply it by the derivative of the inner function.Here is the general form of the chain rule:

• If \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

For our particular problem, we have a composite function involving the inverse hyperbolic tangent and cosine functions.The derivative of \( \tanh^{-1}(u) \) with respect to \( u \) is \( \frac{1}{1-u^2} \).Once you have the derivative of the outer function, find the derivative of the inner function; in this case, \( \frac{d}{dx} \cos(x) = -\sin(x) \).By multiplying these derivatives, you apply the chain rule to get the overall derivative of the composite function.

Inverse Hyperbolic Functions

Inverse hyperbolic functions are the inverse counterparts to the regular hyperbolic functions. They are used in solving many real-world problems involving hyperbolic cosine, sine, and tangent, among others.In this exercise, we encounter the inverse hyperbolic tangent function, denoted as \( \tanh^{-1}(x) \).Its derivative is quite distinct from those of regular functions and is given by:

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

Understanding the derivatives of inverse hyperbolic functions aids greatly in simplifying complex calculus problems.They have important properties and applications, particularly in integration and solving differential equations.When combined with other functions, like in our exercise, it is crucial to clearly separate their derivatives to apply rules like the chain rule accurately.

Trigonometric Derivatives

Trigonometric derivatives pertain to the differentiation of trigonometric functionssuch as sine, cosine, and tangent.They're among the fundamental derivatives you should know in calculus.For our problem, we're interested in the derivative of the cosine function.Remember these basic trigonometric derivatives:

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \( -\sin(x) \).
• The derivative of \( \tan(x) \) is \( \sec^2(x) \).

These derivatives extend to other trigonometric identities as well. Recognizing these basic facts easily guides you when working through complex calculus challenges involving trigonometric functions and helps in deriving solutions to problems like our exercise.By differentiating \( \cos(x) \) in our function, we use its derivative \( -\sin(x) \)to further apply the chain rule, ultimately simplifying even further to expressions like \( -\csc(x) \).