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Question 17 If h (z) = (3z find h'(z) (3z – 15)e¹³z,

Question 17 If h (z) = (3z find h'(z) (3z – 15)e¹³z,

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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### Question 17 If \( h(z) = (3z - 15)e^{13z} \), find \( h'(z) = \) --- #### Explanation The question asks you to find the derivative \( h'(z) \) of the function \( h(z) = (3z - 15)e^{13z} \). This involves using the product rule and the chain rule for differentiation. **Steps to find \( h'(z) \):** 1. **Identify the functions:** - \( u(z) = 3z - 15 \) - \( v(z) = e^{13z} \) 2. **Derivatives of components:** - \( u'(z) = 3 \) - \( v'(z) = 13e^{13z} \) (using the chain rule) 3. **Apply the Product Rule:** - The product rule is given by \( (uv)' = u'v + uv' \). - Substitute back into the formula: \[ h'(z) = u'(z)v(z) + u(z)v'(z) = (3)e^{13z} + (3z - 15)(13e^{13z}) \] 4. **Simplify:** - Combine and factor terms as needed. --- This explanation and format would be appropriate for an educational website looking to teach the product rule and chain rule.
Transcribed Image Text:### Question 17 If \( h(z) = (3z - 15)e^{13z} \), find \( h'(z) = \) --- #### Explanation The question asks you to find the derivative \( h'(z) \) of the function \( h(z) = (3z - 15)e^{13z} \). This involves using the product rule and the chain rule for differentiation. **Steps to find \( h'(z) \):** 1. **Identify the functions:** - \( u(z) = 3z - 15 \) - \( v(z) = e^{13z} \) 2. **Derivatives of components:** - \( u'(z) = 3 \) - \( v'(z) = 13e^{13z} \) (using the chain rule) 3. **Apply the Product Rule:** - The product rule is given by \( (uv)' = u'v + uv' \). - Substitute back into the formula: \[ h'(z) = u'(z)v(z) + u(z)v'(z) = (3)e^{13z} + (3z - 15)(13e^{13z}) \] 4. **Simplify:** - Combine and factor terms as needed. --- This explanation and format would be appropriate for an educational website looking to teach the product rule and chain rule.
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