# Multi variable derivative

1. Jun 19, 2008

### snoggerT

whats derivative of arctan (xy) with respect to x

3. The attempt at a solution

- I feel kind of stupid for this one, but I can't seem to figure out the derivative here. I know with one variable that arctan goes to 1/(1+x^2), but I'm not sure what to do with two variables. please help.

2. Jun 19, 2008

### konthelion

Since you know that $$\frac{d}{dx} \arctan x = \frac{1}{1+x^2}$$, you have to apply the chain rule.

Last edited: Jun 19, 2008
3. Jun 19, 2008

### Defennder

The more general formulae would be $$\frac{d}{dx} \arctan f(x) = \frac{f'(x)}{1+f(x)^2}$$. You have to use implicit differentiation and the chain rule to do this one.

4. Jun 19, 2008

### snoggerT

- I'm still not grasping it. We haven't really done any problems with implicit differentiation like this.

5. Jun 19, 2008

### rocomath

$$y'=\frac{dy}{dx}$$

$$y=\tan^{-1}{(xy)} \rightarrow y=\tan^{-1}f(x)$$

Let $$f(x)=xy \rightarrow f'(x)=xy'+y$$

$$y'=\frac{f'(x)}{1+[f(x)]^2}$$

Now you take it.

Last edited: Jun 19, 2008
6. Jun 19, 2008

### rootX

try finding derv of
arctan (5x)
or
arctan(10x)
..
or
arctan(a.x)
..
now substitute your y for a

7. Jun 20, 2008

### tiny-tim

Or d(arctan(xy))/dx = d(arctan(xy))/d(xy) times (d(xy)/dx).