Textbook Question
Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.
y =√x³+x−1 at y=3
167
views
6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Inverse Trigonometric Functions
2:50 minutesProblem 7.R.17
Textbook Question
10–19. Derivatives Find the derivatives of the following functions.
f(x) = tanh⁻¹(cos x)
Verified step by step guidance1
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.
Recall the derivative of the inverse hyperbolic tangent function: \(\frac{d}{du} \tanh^{-1}(u) = \frac{1}{1 - u^2}\), valid for \(|u| < 1\).
Apply the chain rule: \(\frac{d}{dx} \tanh^{-1}(\cos x) = \frac{1}{1 - (\cos x)^2} \cdot \frac{d}{dx}(\cos x)\).
Find the derivative of the inner function: \(\frac{d}{dx}(\cos x) = -\sin x\).
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.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
3:17
Inverse Tangent
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.
Recommended video:
05:02Intro to the Chain Rule
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.
Recommended video:
03:53Derivatives of Sine & Cosine
Related Videos
Related Practice