10–19. Derivatives Find the derivatives of the following functions.
f(x) = tanh⁻¹(cos x)

Verified step by step guidance

1. Recognize that the function is a composition of two functions: the inverse hyperbolic tangent function, \(\tanh^{-1}(u)\), where \(u = \cos x\). We will need to use the chain rule to differentiate it.
2. Recall the derivative of the inverse hyperbolic tangent function: \(\frac{d}{du} \tanh^{-1}(u) = \frac{1}{1 - u^2}\), valid for \(|u| < 1\).
3. Apply the chain rule: \(\frac{d}{dx} \tanh^{-1}(\cos x) = \frac{1}{1 - (\cos x)^2} \cdot \frac{d}{dx}(\cos x)\).
4. Find the derivative of the inner function: \(\frac{d}{dx}(\cos x) = -\sin x\).
5. Combine the results to write the derivative as \(f'(x) = \frac{-\sin x}{1 - \cos^2 x}\). You can simplify the denominator using the Pythagorean identity if desired.

Key Concepts

Inverse Hyperbolic Tangent Function (tanh⁻¹)

The inverse hyperbolic tangent function, tanh⁻¹(x), returns the value whose hyperbolic tangent is x. It is defined for |x| < 1 and has the derivative d/dx [tanh⁻¹(x)] = 1 / (1 - x²). Understanding its domain and derivative formula is essential for differentiating compositions involving tanh⁻¹.

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that the derivative of f(g(x)) is f'(g(x)) multiplied by g'(x). Applying the chain rule correctly allows us to differentiate tanh⁻¹(cos x) by combining the derivative of tanh⁻¹ with that of cos x.

Derivative of Cosine Function

The cosine function, cos x, is a basic trigonometric function whose derivative is -sin x. Knowing this derivative is crucial when applying the chain rule to functions like tanh⁻¹(cos x), as it provides the inner function's rate of change needed for the overall derivative.