3.5(1) Q7) Hey! I need help with the following question, thank you in advance! ### Using Implicit Differentiation to Find the Tangent Line to a CurveImplicit differentiation is a useful tool for finding the equation of tangent lines to curves that may not be easily expressed in the form \( y = f(x) \). In this example, we will find the equation of the tangent line to a given ellipse.#### Problem Statement:Given the equation of an ellipse:\[ 3x^2 + xy + 3y^2 = 7 \]Find the equation of the tangent line to the curve at the point \( (1, 1) \).#### Steps:1. **Implicit Differentiation:** Differentiate both sides of the equation with respect to \( x \) while treating \( y \) as an implicit function of \( x \): \[ \frac{d}{dx}(3x^2) + \frac{d}{dx}(xy) + \frac{d}{dx}(3y^2) = \frac{d}{dx}(7) \]2. **Solve for \(\frac{dy}{dx}\):** Use the chain rule and product rule where necessary.3. **Substitute the Point \((1, 1)\):** Evaluate the differentiated equation at the point to find the slope of the tangent line.4. **Tangent Line Equation:** Use the point-slope form of the equation of a line to find the specific equation of the tangent line.#### Example Calculation:The resultant equation after implicit differentiation will be:\[ 6x + y + x\frac{dy}{dx} + 6y\frac{dy}{dx} = 0 \]Substitute \( x = 1 \) and \( y = 1 \):\[ 6(1) + 1 + 1\frac{dy}{dx} + 6(1)\frac{dy}{dx} = 0 \]Solve for \( \frac{dy}{dx} \):\[ 6 + 1 + (\frac{dy}{dx})(1 + 6) = 0 \]\[ 7 + 7\frac{dy}{dx} = 0 \]\[ \frac{dy}{dx} = -1 \]The slope of the tangent line at \( (1, 1) \) is \(-1\).Using the point-slope form \( y - y_1 = m(x - x_

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Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 3x² + xy + 3y² = 7, (1, 1) (ellipse) y =

Calculus: Early Transcendentals

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ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

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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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3.5(1) Q7) Hey! I need help with the following question, thank you in advance!

### Using Implicit Differentiation to Find the Tangent Line to a Curve

Implicit differentiation is a useful tool for finding the equation of tangent lines to curves that may not be easily expressed in the form \( y = f(x) \). In this example, we will find the equation of the tangent line to a given ellipse.

#### Problem Statement:
Given the equation of an ellipse:
\[ 3x^2 + xy + 3y^2 = 7 \]
Find the equation of the tangent line to the curve at the point \( (1, 1) \).

#### Steps:
1. **Implicit Differentiation:** Differentiate both sides of the equation with respect to \( x \) while treating \( y \) as an implicit function of \( x \):
\[ \frac{d}{dx}(3x^2) + \frac{d}{dx}(xy) + \frac{d}{dx}(3y^2) = \frac{d}{dx}(7) \]

2. **Solve for \(\frac{dy}{dx}\):** Use the chain rule and product rule where necessary.

3. **Substitute the Point \((1, 1)\):** Evaluate the differentiated equation at the point to find the slope of the tangent line.

4. **Tangent Line Equation:** Use the point-slope form of the equation of a line to find the specific equation of the tangent line.

#### Example Calculation:
The resultant equation after implicit differentiation will be:
\[ 6x + y + x\frac{dy}{dx} + 6y\frac{dy}{dx} = 0 \]

Substitute \( x = 1 \) and \( y = 1 \):
\[ 6(1) + 1 + 1\frac{dy}{dx} + 6(1)\frac{dy}{dx} = 0 \]

Solve for \( \frac{dy}{dx} \):
\[ 6 + 1 + (\frac{dy}{dx})(1 + 6) = 0 \]
\[ 7 + 7\frac{dy}{dx} = 0 \]
\[ \frac{dy}{dx} = -1 \]

The slope of the tangent line at \( (1, 1) \) is \(-1\).

Using the point-slope form \( y - y_1 = m(x - x_

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