Question

In the xy-plane, the line tangent to the graph of x² + xy + y = 3 at the point (1, 1) has a slope of​

Answer 1

The slope of the tangent line is -3/2.

Explanation:

The slope of a tangent line is given by the derivative evaluated at the point of tangency.

To find the derivative dy/dx, use implicit differentiation.

The derivative of the first term is 2x.

The derivative of the second term is found by using the Product Rule.

The derivative of y is dy/dx.

The derivative of 3 is 0.

Differentiating each term produces

`2x + x\cdot(dy)/(dx)+y\cdot 1+(dy)/(dx) = 0`

Solve for dy/dx.

`2x+y+(x+1)(dy)/(dx)=0 \\(dy)/(dx)=(-2x-y)/(x+1)`

Plug in the point (1, 1).

`(dy)/(dx)=(-2-1)/(1+1) =-(3)/(2)`