Textbook Question
Theorem 7.8
Differentiate sinh⁻¹ x = ln (x + √(x² + 1)) to show that d/dx (sinh⁻¹ x) = 1 / √(x² + 1).
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6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Exponential & Logarithmic Functions
2:27 minutesProblem 7.2.14
Textbook Question
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
14. y = ln(2θ+2)
Verified step by step guidance1
Identify the function given: \(y = \ln(2\theta + 2)\), where \(y\) is expressed in terms of \(\theta\).
Recall the derivative rule for the natural logarithm function: if \(y = \ln(u)\), then \(\frac{dy}{d\theta} = \frac{1}{u} \cdot \frac{du}{d\theta}\).
Set \(u = 2\theta + 2\) and find its derivative with respect to \(\theta\): \(\frac{du}{d\theta} = 2\).
Apply the chain rule by substituting \(u\) and \(\frac{du}{d\theta}\) into the derivative formula: \(\frac{dy}{d\theta} = \frac{1}{2\theta + 2} \cdot 2\).
Simplify the expression if possible to write the derivative in its simplest form.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of the Natural Logarithm Function
The derivative of ln(u), where u is a differentiable function of a variable, is given by (1/u) times the derivative of u. This rule allows us to differentiate logarithmic functions by applying the chain rule to the inner function.
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Derivative of the Natural Logarithmic Function
Chain Rule
The chain rule is used to differentiate composite functions. It states that the derivative of a function composed of another function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
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05:02Intro to the Chain Rule
Differentiation with Respect to a Variable
When differentiating, it is important to identify the variable with respect to which the derivative is taken. In this problem, the variable is θ, so all derivatives must be computed considering θ as the independent variable.
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07:15Separation of Variables
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