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Use the Chain Rule to evaluate the partial derivative at the point (q, r) = (2, 3), where h(u, v) = ue", u = q*, v = qr². да (Use symbolic notation and fractions where needed.) dh aq (gr) Incorrect = 176e16

Use the Chain Rule to evaluate the partial derivative at the point (q, r) = (2, 3), where h(u, v) = ue", u = q*, v = qr². да (Use symbolic notation and fractions where needed.) dh aq (gr) Incorrect = 176e16

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter3: The Derivative
Section3.5: Graphical Differentiation
Problem 2E
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**Title: Using the Chain Rule to Evaluate Partial Derivatives** **Objective:** Learn how to utilize the Chain Rule to calculate the partial derivative of a function at given points. **Problem Statement:** Use the Chain Rule to evaluate the partial derivative \(\frac{\partial h}{\partial q}\) at the point \((q, r) = (2, 3)\), where \( h(u, v) = ue^v \), \( u = q^4 \), and \( v = qr^2 \). *(Use symbolic notation and fractions where needed.)* **Solution Process:** Given: - \( h(u, v) = ue^v \) - \( u = q^4 \) - \( v = qr^2 \) We need to find \(\frac{\partial h}{\partial q}\) at \((q, r) = (2, 3)\). **Steps:** 1. **Compute partial derivatives of \( h \) with respect to \( u \) and \( v \):** \[ \frac{\partial h}{\partial u} = e^v \] \[ \frac{\partial h}{\partial v} = ue^v \] 2. **Calculate partial derivatives of \( u \) and \( v \) with respect to \( q \):** \[ \frac{\partial u}{\partial q} = 4q^3 \] \[ \frac{\partial v}{\partial q} = r^2 \] 3. **Apply the Chain Rule:** \[ \frac{\partial h}{\partial q} = \frac{\partial h}{\partial u} \frac{\partial u}{\partial q} + \frac{\partial h}{\partial v} \frac{\partial v}{\partial q} \] 4. **Substitute the partial derivatives into the Chain Rule formula:** \[ \frac{\partial h}{\partial q} = (e^v)(4q^3) + (ue^v)(r^2) \] 5. **Substitute \( u \) and \( v \) using \( u = q^4 \) and \( v = qr^2 \):** \[ \frac{\partial h}{\partial q} = (e
Transcribed Image Text:**Title: Using the Chain Rule to Evaluate Partial Derivatives** **Objective:** Learn how to utilize the Chain Rule to calculate the partial derivative of a function at given points. **Problem Statement:** Use the Chain Rule to evaluate the partial derivative \(\frac{\partial h}{\partial q}\) at the point \((q, r) = (2, 3)\), where \( h(u, v) = ue^v \), \( u = q^4 \), and \( v = qr^2 \). *(Use symbolic notation and fractions where needed.)* **Solution Process:** Given: - \( h(u, v) = ue^v \) - \( u = q^4 \) - \( v = qr^2 \) We need to find \(\frac{\partial h}{\partial q}\) at \((q, r) = (2, 3)\). **Steps:** 1. **Compute partial derivatives of \( h \) with respect to \( u \) and \( v \):** \[ \frac{\partial h}{\partial u} = e^v \] \[ \frac{\partial h}{\partial v} = ue^v \] 2. **Calculate partial derivatives of \( u \) and \( v \) with respect to \( q \):** \[ \frac{\partial u}{\partial q} = 4q^3 \] \[ \frac{\partial v}{\partial q} = r^2 \] 3. **Apply the Chain Rule:** \[ \frac{\partial h}{\partial q} = \frac{\partial h}{\partial u} \frac{\partial u}{\partial q} + \frac{\partial h}{\partial v} \frac{\partial v}{\partial q} \] 4. **Substitute the partial derivatives into the Chain Rule formula:** \[ \frac{\partial h}{\partial q} = (e^v)(4q^3) + (ue^v)(r^2) \] 5. **Substitute \( u \) and \( v \) using \( u = q^4 \) and \( v = qr^2 \):** \[ \frac{\partial h}{\partial q} = (e
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