Tangent line off a curve

Question

I'm having trouble figuring out how to find the equation to a tangent line off a curve. I know you use implicit differentiation of the function, but lets say the derivative found is still a difficult function, such as $-2x + y^2/-2xy + 1$ Now, the formula for the tangent line is y = mx + c and lets say it passes through the point (-3, -5). How would you go about figuring this out?

Answer 1

Firstly you find $\frac {dy}{dx}$. For this you only need to use implicit differentiation if you do not have an explicit formula for $y$. Then substitute the coordinates of the point of tangency into your expression for $\frac {dy}{dx}$.

Example $\frac {-2x+y^2}{-2xy+1}=\frac {-31}{29}$ at $(-3,-5)$.

This gradient is the $"m"$ in $y=mx+c$.

Example $y=\frac {-31}{29}x+c$. You then use the point $(-3,-5)$ to find $"c"$.