Question

using implicit differentiation find y 'for x2-3xy-4y2=23 and find the equation of the tangent line of the graph at (-1,2)

Answer 1

The point of implicit differentiation is to get y' when it isn't possible to get y as a function of x. So, when we implicitly differentiate, we differentiate both sides with respect to x because we want y as a function of x. So:

I'm assuming x2 = x² and y2 = y²

\[x^{2} - 3xy - 4y^{2} = 23\]
\[d/dx(x^{2} - 3xy - 4y^{2})=d/dx(23)\]
\[2x - d/dx(3xy) - 8y(dy/dx)=0\]
Notice that the middle term is a product rule so and the product rule is the derivative of the first fuction * the second function + the first function * the derivative of the second function.
\[d/dx(3xy) = 3y + 3x(dy/dx)\]

So the equation is:
\[2x - (3y + 3x(dy/dx)) - 8y(dy/dx) = 0\]

Simplify:
\[2x - 3y - 3x(dy/dx) - 8y(dy/dx) = 0\]
\[2x - 3y = (3x + 8y)(dy/dx)\]
\[dy/dx = m= (2x - 3y) / (3x + 8y)\]
The slope of the tangent line is the derivative so just plug in the point (-1,2) in for x and y in the above equation and then use the same point to get it in y = mx + b form:
\[y = (-8/13)x + b\]
Now solve this equation for b using the point (-1,2) again and you'll get the equation for the tangent line.