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Find the derivative of f (x) = (x8 – 5x) (x² f(x)= + 10x 10).

Find the derivative of f (x) = (x8 – 5x) (x² f(x)= + 10x 10).

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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To find the derivative of the function \( f(x) = \left( x^8 - 5x \right) \left( x^2 + 10x - 10 \right) \), you can use the product rule. The product rule states that if you have a function \( f(x) = g(x) \cdot h(x) \), then the derivative \( f'(x) \) is given by: \[ f'(x) = g'(x)h(x) + g(x)h'(x). \] Here, let \( g(x) = x^8 - 5x \) and \( h(x) = x^2 + 10x - 10 \). First, compute the derivatives of \( g(x) \) and \( h(x) \): \[ g'(x) = \frac{d}{dx}(x^8 - 5x) = 8x^7 - 5 \] \[ h'(x) = \frac{d}{dx}(x^2 + 10x - 10) = 2x + 10 \] Now apply the product rule: \[ f'(x) = (8x^7 - 5)(x^2 + 10x - 10) + (x^8 - 5x)(2x + 10) \] The given image also includes an empty answer box followed by two icons, which likely represent actions to check or submit the answer. Here is how you might transcribe this information into the educational website: --- **Find the derivative of \( f(x) = \left( x^8 - 5x \right) \left( x^2 + 10x - 10 \right) \).** \[ f'(x) = \boxed{\hspace{5cm}} \quad \underset{\text{Check}}{\rightarrow} \] By applying the product rule, we have: \[ f'(x) = (8x^7 - 5)(x^2 + 10x - 10) + (x^8 - 5x)(2x + 10) \]
Transcribed Image Text:To find the derivative of the function \( f(x) = \left( x^8 - 5x \right) \left( x^2 + 10x - 10 \right) \), you can use the product rule. The product rule states that if you have a function \( f(x) = g(x) \cdot h(x) \), then the derivative \( f'(x) \) is given by: \[ f'(x) = g'(x)h(x) + g(x)h'(x). \] Here, let \( g(x) = x^8 - 5x \) and \( h(x) = x^2 + 10x - 10 \). First, compute the derivatives of \( g(x) \) and \( h(x) \): \[ g'(x) = \frac{d}{dx}(x^8 - 5x) = 8x^7 - 5 \] \[ h'(x) = \frac{d}{dx}(x^2 + 10x - 10) = 2x + 10 \] Now apply the product rule: \[ f'(x) = (8x^7 - 5)(x^2 + 10x - 10) + (x^8 - 5x)(2x + 10) \] The given image also includes an empty answer box followed by two icons, which likely represent actions to check or submit the answer. Here is how you might transcribe this information into the educational website: --- **Find the derivative of \( f(x) = \left( x^8 - 5x \right) \left( x^2 + 10x - 10 \right) \).** \[ f'(x) = \boxed{\hspace{5cm}} \quad \underset{\text{Check}}{\rightarrow} \] By applying the product rule, we have: \[ f'(x) = (8x^7 - 5)(x^2 + 10x - 10) + (x^8 - 5x)(2x + 10) \]
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