# Implicit differentiation

• Apr 26th 2010, 05:19 PM
Pandora
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.

• Apr 26th 2010, 05:27 PM
dwsmith
Quote:

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.

$\frac{dy}{dx}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}$
• Apr 26th 2010, 05:37 PM
Pandora
Thank you. Would you mind working it out from the facts? I am pretty horrible at implicit differentiation.
• Apr 26th 2010, 05:41 PM
dwsmith
Quote:

Originally Posted by Pandora
Thank you. Would you mind working it out from the facts? I am pretty horrible at implicit differentiation.

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

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

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

Now simplify and solve.