Thread: Implicit differentiation

1. Implicit differentiation

I was wondering if someone could help me out.
I was given the function f(x,y)=squareroot(1+ x^2 + y^2) and asked to find the slope of a tangent to a level curve at point (-2,2).

I have managed to work out the gradient of f at (-2,2) to be (-2/3, 2/3), and I know it has to be orthogonal to the tangent, but I am unsure as to how to work out the slope of the tangent.

Would appreciate your help

2. Originally Posted by Pandora
I was wondering if someone could help me out.
I was given the function f(x,y)=squareroot(1+ x^2 + y^2) and asked to find the slope of a tangent to a level curve at point (-2,2).

I have managed to work out the gradient of f at (-2,2) to be (-2/3, 2/3), and I know it has to be orthogonal to the tangent, but I am unsure as to how to work out the slope of the tangent.

Would appreciate your help
$\displaystyle \frac{dy}{dx}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}$

3. Thank you. Would you mind working it out from the facts? I am pretty horrible at implicit differentiation.

4. Originally Posted by Pandora
Thank you. Would you mind working it out from the facts? I am pretty horrible at implicit differentiation.
$\displaystyle \frac{\partial F}{\partial x}=\frac{x}{\sqrt{1+x^2+y^2}}$

$\displaystyle \frac{\partial F}{\partial y}=\frac{y}{\sqrt{1+x^2+y^2}}$

$\displaystyle \frac{dy}{dx}=-\frac{\frac{x}{\sqrt{1+x^2+y^2}}}{\frac{y}{\sqrt{1 +x^2+y^2}}}$

Now simplify and solve.