Question

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x²−7xy=8(1,−1)

Answer 1

To find the equation of the tangent line to the curve x²−7xy=8 at the point (1, -1), we use implicit differentiation and the point-slope formula. The equation of the tangent line is y = (9/7)x - (16/7).

Step-by-step explanation:

To find the equation of the tangent line to the curve, we will use implicit differentiation. Starting with the equation: x²−7xy=8

First, we differentiate both sides of the equation with respect to x:

2x - 7x(dy/dx) - 7y = 0

Next, we need to find the derivative dy/dx. Rearranging the equation, we get:

dy/dx = (2x - 7y) / (7x)

Now, we substitute the coordinates of the given point (1, -1) into the derivative equation to find the slope at that point:

dy/dx = (2(1) - 7(-1)) / (7(1)) = 9/7

Therefore, the slope of the tangent line at the point (1, -1) is 9/7. Using the point-slope form of a linear equation, we can write the equation of the tangent line as: y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point. Plugging in the values, we get:

y - (-1) = (9/7)(x - 1)

Simplifying the equation gives:

y = (9/7)x - (16/7)

Answer 2

The equation of the tangent line to the curve
`\(x^2 - 7xy = 8\)` at the point
`\((1, -1)\) is \(y = (9)/(7)x - (16)/(7)\).`

Implicit differentiation involves differentiating both sides of an equation with respect to x (or y) treating one variable as a function of the other.

Given the equation
`\(x^2 - 7xy = 8\)`, we'll differentiate both sides with respect to x to find
`\((dy)/(dx)\).`

`\[(d)/(dx) (x^2 - 7xy) = (d)/(dx) 8\]`

Differentiating term by term:

`\[(d)/(dx) (x^2) - (d)/(dx) (7xy) = 0\]`

`\[2x - 7(dy)/(dx) \cdot x - 7y = 0\]`

Now, let's solve this equation for
`\((dy)/(dx)\):`

`\[(dy)/(dx) = (2x - 7y)/(7x)\]`

To find the slope of the tangent line at the point (1, -1), substitute x = 1 and y = -1 into
`\((dy)/(dx)\)`:

`\[(dy)/(dx) \Bigg|_((1,-1)) = (2(1) - 7(-1))/(7(1)) = (2 + 7)/(7) = (9)/(7)\]`

Now that we have the slope of the tangent line, we'll use the point-slope form of the equation of a line to find the equation of the tangent line:

`\[y - y_1 = m(x - x_1)\]`

Given the point (1, -1) and the slope
`\(m = (9)/(7)\)`, plug these values into the equation:

`\[y - (-1) = (9)/(7)(x - 1)\]`

`\[y + 1 = (9)/(7)x - (9)/(7)\]`

`\[y = (9)/(7)x - (16)/(7)\]`