Textbook Question
Find the derivative of the inverse of the following functions. Express the result with x as the independent variable.
f(x) = x^-1/3
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6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Inverse Trigonometric Functions
2:40 minutesProblem 7.R.12
Textbook Question
10–19. Derivatives Find the derivatives of the following functions.
f(x) = (sinh x) / (1 + sinh x)
Verified step by step guidance1
Identify the function to differentiate: \(f(x) = \frac{\sinh x}{1 + \sinh x}\), which is a quotient of two functions.
Recall the Quotient Rule for derivatives: if \(f(x) = \frac{u(x)}{v(x)}\), then \(f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\).
Set \(u(x) = \sinh x\) and \(v(x) = 1 + \sinh x\). Compute their derivatives: \(u'(x) = \cosh x\) and \(v'(x) = \cosh x\).
Apply the Quotient Rule: substitute \(u\), \(v\), \(u'\), and \(v'\) into the formula to get \(f'(x) = \frac{\cosh x (1 + \sinh x) - \sinh x \cosh x}{(1 + \sinh x)^2}\).
Simplify the numerator by factoring and combining like terms to express the derivative in its simplest form.
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.
Hyperbolic Functions
Hyperbolic functions like sinh(x) are analogs of trigonometric functions but based on hyperbolas. The sinh function is defined as (e^x - e^(-x))/2 and has properties similar to sine, including specific derivatives that are essential for differentiation.
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Quotient Rule
The quotient rule is used to differentiate functions expressed as one function divided by another. It states that the derivative of f(x)/g(x) is (f'(x)g(x) - f(x)g'(x)) / [g(x)]^2, which is crucial for finding the derivative of the given function.
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Derivative of Hyperbolic Sine
The derivative of sinh(x) is cosh(x), another hyperbolic function defined as (e^x + e^(-x))/2. Knowing this derivative is necessary to apply the quotient rule correctly when differentiating the given function.
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