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Differentiate the function. y = 1 (5x – 16)2

Differentiate the function. y = 1 (5x – 16)2

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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 please show your derivative by hand and simplify

**Differentiation of the Function** To differentiate the function given below: \[ y = \frac{1}{(5x - 16)^2} \] Apply the chain rule and the power rule for derivatives. **Solution:** Given the function: \[ y = \frac{1}{(5x - 16)^2} = (5x - 16)^{-2} \] Differentiate with respect to \( x \): 1. Use the power rule: \(\frac{d}{dx}[u^n] = n \cdot u^{n-1} \cdot \frac{du}{dx}\) 2. Set \( u = 5x - 16 \); then \(\frac{du}{dx} = 5\). 3. Applying the power rule: \[ \frac{dy}{dx} = -2 \cdot (5x - 16)^{-3} \cdot \frac{d}{dx}(5x - 16) \] \[ \frac{dy}{dx} = -2 \cdot (5x - 16)^{-3} \cdot 5 \] \[ \frac{dy}{dx} = -10 \cdot (5x - 16)^{-3} \] The image presents the final derivative value as: \[ \frac{dy}{dx} = 5 \] Note: The provided derivative value of 5 seems to be incorrect given the differentiation steps above. Verify or recalculate the provided value to ensure accuracy in instructional materials.
Transcribed Image Text:**Differentiation of the Function** To differentiate the function given below: \[ y = \frac{1}{(5x - 16)^2} \] Apply the chain rule and the power rule for derivatives. **Solution:** Given the function: \[ y = \frac{1}{(5x - 16)^2} = (5x - 16)^{-2} \] Differentiate with respect to \( x \): 1. Use the power rule: \(\frac{d}{dx}[u^n] = n \cdot u^{n-1} \cdot \frac{du}{dx}\) 2. Set \( u = 5x - 16 \); then \(\frac{du}{dx} = 5\). 3. Applying the power rule: \[ \frac{dy}{dx} = -2 \cdot (5x - 16)^{-3} \cdot \frac{d}{dx}(5x - 16) \] \[ \frac{dy}{dx} = -2 \cdot (5x - 16)^{-3} \cdot 5 \] \[ \frac{dy}{dx} = -10 \cdot (5x - 16)^{-3} \] The image presents the final derivative value as: \[ \frac{dy}{dx} = 5 \] Note: The provided derivative value of 5 seems to be incorrect given the differentiation steps above. Verify or recalculate the provided value to ensure accuracy in instructional materials.
Expert Solution
Step 1: Given value of function y

Given, y equals 1 over left parenthesis 5 x minus 16 right parenthesis squared


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