1. ## Tangent plane help.

I'm trying to understand the computation of tangent planes. Mainly my problem is when to use the formula that involves the gradient. For example here are two different questions.

Find an equation of the tangent plane z = f(x,y) at the point (2,3, f(2,3)) at:

f(x,y) = (x)/[x^2+y^2]

Find the equation of the tangent plane at the point (-3,1,0) to the graph of z = f(x,y) defined implicitly by x(y^2+z^2) + ye^(xz) = -2

in the first question this can be solved by
fx(x0y0)(x−x0)+fy(x0y0)(y−y0)−(z−z0) = 0

the second question requires the use of the gradient to solve. I just want to know where to use each formula and why?

2. ## Re: Tangent plane help.

Originally Posted by Kuma
I'm trying to understand the computation of tangent planes. Mainly my problem is when to use the formula that involves the gradient. For example here are two different questions.

Find an equation of the tangent plane z = f(x,y) at the point (2,3, f(2,3)) at:

f(x,y) = (x)/[x^2+y^2]

Find the equation of the tangent plane at the point (-3,1,0) to the graph of z = f(x,y) defined implicitly by x(y^2+z^2) + ye^(xz) = -2

in the first question this can be solved by
fx(x0y0)(x−x0)+fy(x0y0)(y−y0)−(z−z0) = 0

the second question requires the use of the gradient to solve. I just want to know where to use each formula and why?
The point of tangentcy is $\displaystyle \displaystyle \left(2, 3, \frac{2}{13}\right)$, and the function is $\displaystyle \displaystyle z = \frac{x}{x^2 + y^2} \implies \frac{x}{x^2 + y^2} - z = 0$

The normal vector is $\displaystyle \displaystyle \nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$, which you need to evaluate at the point $\displaystyle \displaystyle \left(2, 3, \frac{2}{13}\right)$

The tangent plane $\displaystyle \displaystyle ax + by + cz = d$ has the same coefficients as the normal vector.