Find dx dx || by implicit differentiation. 6x² + 7xy - y² = 8

Calculus: Early Transcendentals
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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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What are the basics of Implicit Differentiation in Calculus and how would you solve these probelms?

**Problem Statement:**

Find \(\frac{dy}{dx}\) by implicit differentiation.

Given the equation:

\[ 6x^2 + 7xy - y^2 = 8 \]

**Solution:**

Calculate \(\frac{dy}{dx}\) and fill the box provided with your answer. 

**Approach:**

1. Differentiate both sides of the equation with respect to \(x\).
2. Apply the product rule to the term involving \(xy\).
3. Solve for \(\frac{dy}{dx}\).

**Note:**

Implicit differentiation is used when \(y\) is defined implicitly as a function of \(x\).
Transcribed Image Text:**Problem Statement:** Find \(\frac{dy}{dx}\) by implicit differentiation. Given the equation: \[ 6x^2 + 7xy - y^2 = 8 \] **Solution:** Calculate \(\frac{dy}{dx}\) and fill the box provided with your answer. **Approach:** 1. Differentiate both sides of the equation with respect to \(x\). 2. Apply the product rule to the term involving \(xy\). 3. Solve for \(\frac{dy}{dx}\). **Note:** Implicit differentiation is used when \(y\) is defined implicitly as a function of \(x\).
**Problem: Implicit Differentiation**

Find \(\frac{dy}{dx}\) by implicit differentiation.

Given equation:

\[ e^{x/y} = 9x - y \]

Calculate \(\frac{dy}{dx}\):

\[\frac{dy}{dx} = \boxed{\phantom{answer}}\]

**Explanation:**

The task is to find the derivative \(\frac{dy}{dx}\) of the given equation implicitly. This involves differentiating both sides of the equation with respect to \(x\), making use of the chain rule and recognizing that \(y\) is a function of \(x\).
Transcribed Image Text:**Problem: Implicit Differentiation** Find \(\frac{dy}{dx}\) by implicit differentiation. Given equation: \[ e^{x/y} = 9x - y \] Calculate \(\frac{dy}{dx}\): \[\frac{dy}{dx} = \boxed{\phantom{answer}}\] **Explanation:** The task is to find the derivative \(\frac{dy}{dx}\) of the given equation implicitly. This involves differentiating both sides of the equation with respect to \(x\), making use of the chain rule and recognizing that \(y\) is a function of \(x\).
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