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Jsing only exponential (e*), cubic (x³), and the cosine (cos x) functions, create anot unction for which the derivative computation requires using Chain Rule twice. Compu lerivative.

Jsing only exponential (e*), cubic (x³), and the cosine (cos x) functions, create anot unction for which the derivative computation requires using Chain Rule twice. Compu lerivative.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Exercise: Derivative Computation Using the Chain Rule** **Task:** Using only exponential \((e^x)\), cubic \((x^3)\), and cosine \((\cos x)\) functions, create another function for which the derivative computation requires using the Chain Rule twice. Compute that derivative. **Solution Steps:** 1. **Create a Composite Function:** - Form a function \( f(x) = e^{\cos(x^3)} \). - This function is created using the exponential function \(e^x\), the cosine function \(\cos x\), and the cubic function \(x^3\). 2. **Apply the Chain Rule Twice to Differentiate:** - **Outer Function:** \( e^u \), where \( u = \cos(v) \) and \( v = x^3 \). - **Inner Function 1:** \( \cos(v) \). - **Inner Function 2:** \( x^3 \). 3. **Differentiate Step-by-Step:** - \( \frac{d}{dx} e^{\cos(x^3)} = e^{\cos(x^3)} \cdot \frac{d}{dx}(\cos(x^3)) \). - Apply the Chain Rule to \( \cos(x^3) \): - \( \frac{d}{dx}\cos(x^3) = -\sin(x^3) \cdot \frac{d}{dx}(x^3) \). - Differentiate \( x^3 \): - \( \frac{d}{dx}(x^3) = 3x^2 \). 4. **Combine the Results:** - \( \frac{d}{dx} e^{\cos(x^3)} = e^{\cos(x^3)} \cdot (-\sin(x^3)) \cdot 3x^2 \). - Simplify: - \( = -3x^2 \sin(x^3) e^{\cos(x^3)} \). The derivative is: \(-3x^2 \sin(x^3) e^{\cos(x^3)}\).
Transcribed Image Text:**Exercise: Derivative Computation Using the Chain Rule** **Task:** Using only exponential \((e^x)\), cubic \((x^3)\), and cosine \((\cos x)\) functions, create another function for which the derivative computation requires using the Chain Rule twice. Compute that derivative. **Solution Steps:** 1. **Create a Composite Function:** - Form a function \( f(x) = e^{\cos(x^3)} \). - This function is created using the exponential function \(e^x\), the cosine function \(\cos x\), and the cubic function \(x^3\). 2. **Apply the Chain Rule Twice to Differentiate:** - **Outer Function:** \( e^u \), where \( u = \cos(v) \) and \( v = x^3 \). - **Inner Function 1:** \( \cos(v) \). - **Inner Function 2:** \( x^3 \). 3. **Differentiate Step-by-Step:** - \( \frac{d}{dx} e^{\cos(x^3)} = e^{\cos(x^3)} \cdot \frac{d}{dx}(\cos(x^3)) \). - Apply the Chain Rule to \( \cos(x^3) \): - \( \frac{d}{dx}\cos(x^3) = -\sin(x^3) \cdot \frac{d}{dx}(x^3) \). - Differentiate \( x^3 \): - \( \frac{d}{dx}(x^3) = 3x^2 \). 4. **Combine the Results:** - \( \frac{d}{dx} e^{\cos(x^3)} = e^{\cos(x^3)} \cdot (-\sin(x^3)) \cdot 3x^2 \). - Simplify: - \( = -3x^2 \sin(x^3) e^{\cos(x^3)} \). The derivative is: \(-3x^2 \sin(x^3) e^{\cos(x^3)}\).
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