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Magazine Chapter 8: Problem 4
Find \(f^{\prime}(x)\) if \(f(x)\) is the given expression. \(\sinh ^{-1} e^{x}\)
Short Answer
Expert verified
The derivative is \( f'(x) = \frac{e^x}{\sqrt{e^{2x} + 1}} \).
Step by step solution
01
Understand the Function
The function we have is \( f(x) = \sinh^{-1}(e^x) \), where \( \sinh^{-1}(x) \) is the inverse hyperbolic sine function.
02
Use the Derivative of Inverse Hyperbolic Function
The derivative of the inverse hyperbolic sine, \( \sinh^{-1}(x) \), is given by \( \frac{d}{dx} \sinh^{-1}(x) = \frac{1}{\sqrt{x^2 + 1}} \). We will use this to find the derivative of our function.
03
Apply the Chain Rule
Since \( f(x) = \sinh^{-1}(e^x) \), and \( u = e^x \), the chain rule tells us to first find the derivative with respect to \( u \) and then multiply by the derivative of \( u \) with respect to \( x \).
04
Differentiate with Respect to \( u \)
Compute \( \frac{d}{du} \sinh^{-1}(u) = \frac{1}{\sqrt{u^2 + 1}} \). Substitute \( u = e^x \) into the expression to get \( \frac{1}{\sqrt{(e^x)^2 + 1}} \).
05
Differentiate \( u = e^x \) with Respect to \( x \)
The derivative of \( u = e^x \) is \( \frac{du}{dx} = e^x \).
06
Combine Using the Chain Rule
Now apply the chain rule: \( f'(x) = \frac{d}{du} \sinh^{-1}(u) \cdot \frac{du}{dx} = \frac{1}{\sqrt{(e^x)^2 + 1}} \times e^x \).
07
Simplify the Expression
Simplify the expression \( \frac{e^x}{\sqrt{(e^x)^2 + 1}} \) to get the final derivative: \( f'(x) = \frac{e^x}{\sqrt{e^{2x} + 1}} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Derivatives
Derivatives are a fundamental concept in calculus. They represent how a function changes as its input changes. In essence, the derivative gives the rate of change or the slope of the function at any given point. For a function denoted as \( f(x) \), its derivative is typically represented as \( f'(x) \) or \( \frac{df}{dx} \).
**Why Derivatives Matter**
**Why Derivatives Matter**
- They help in understanding the behavior of functions.
- They are used to find local maxima and minima of functions. This is useful for optimization problems.
- In physics, derivatives are used to determine velocity and acceleration from position-time graphs.
Inverse Hyperbolic Functions
Inverse hyperbolic functions, like \( \sinh^{-1}(x) \), are analogous to trigonometric functions, but they are built from hyperbolic functions. These functions arise in various real-world applications, including calculating distances in hyperbolic geometry and modeling in engineering contexts.
**Understanding \( \sinh^{-1} \)**
**Understanding \( \sinh^{-1} \)**
- \( \sinh^{-1}(x) \) is the inverse of the hyperbolic sine function \( \sinh(x) \).
- The derivative of \( \sinh^{-1}(x) \) is computed using its formula: \( \frac{d}{dx} \sinh^{-1}(x) = \frac{1}{\sqrt{x^2 + 1}} \).
- This derivative tells us how \( \sinh^{-1} \) behaves when \( x \) changes, which is crucial for problems like differentiating \( \sinh^{-1}(e^x) \).
Chain Rule
The chain rule is one of the essential techniques in differentiation. It allows us to find the derivative of composite functions. A composite function involves two or more functions, like \( f(x) = \sinh^{-1}(e^x) \).
**How The Chain Rule Works**
**How The Chain Rule Works**
- It tells us to first differentiate the outer function with respect to the inner function.
- Then, multiply this result by the derivative of the inner function with respect to its variable.
- For example, in \( \sinh^{-1}(e^x) \), the outer function is \( \sinh^{-1} \) and the inner function is \( e^x \).
