Calculating the equation of the tangent plane to $z=f(x,y)$ at $(2,1,\ln(3))$ where $f(x,y)=\ln(\sqrt{x^2+5y^2})$ for $(x,y)\neq (0,0)$

24 Views Asked by At

Let $f(x,y)=\ln(\sqrt{x^2+5y^2})$ for $(x,y)\neq (0,0)$

a) Calculate the partial derivatives $\frac{\partial{f}}{\partial{x}}$ and $\frac{\partial{f}}{\partial{y}}$

b) Calculate the equation of the tangent plane to $z=f(x,y)$ at $(2,1,\ln(3))$

My Approach:

I know how to do the first part, but I don't know how to find part (b).

This is my answer for part (a):

$$\frac{\partial{f}}{\partial{x}}=\frac{1}{\sqrt{x^2+5y^2}}\cdot\frac{1}{2}\cdot \frac{1}{\sqrt{x^2+5y^2}} \cdot2x=\frac{x}{x^2+5y^2}$$

$$\frac{\partial{f}}{\partial{y}}=\frac{5y}{x^2+5y^2}$$

How do I do part b?