Question

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

9x2 + xy + 9y2 = 19, (1, 1) (ellipse)
y = ________________?

Answer 1

To find the equation of the tangent line to the curve, differentiate the given equation implicitly with respect to x, substitute the given point, and use the slope-point form of a line.

Step-by-step explanation:

To find the equation of the tangent line to the curve, we will first differentiate the given equation implicitly with respect to x.

Differentiating 9x^2 + xy + 9y^2 = 19 with respect to x:

• Using the product rule, differentiate the term xy as y + x(dy/dx).
• Differentiate the term 9y^2 as 18yy' using the chain rule.
• The derivative of 19 with respect to x is 0.

Substituting the given point (1, 1) into the equation, we can now solve for the value of dy/dx, which represents the slope of the tangent line. Finally, we can use the slope-point form of a line to determine the equation of the tangent line.

The equation of the tangent line is y = 3x - 6.