Question

What is the derivative of `7xy`?

Answer 1

That depends on what variable you want to take the derivative with respect to. With respect to `x` the answer is `7*y`, while with respect to `y` the answer is `7*x`.

When you take a derivative of a function you get an expression that represents the rate of change or slope of that function. With single variable calculus you usually only ever take derivatives for a function with respect to the independent variable. This is usually represented by `x`. However, with multivariate calculus you often times have a function defined in three or more dimensions. For instance the above function might look something like this.

`z=7*x*y`

You could then choose to take the derivative with respect to either the `x` axis or the `y` axis. If you take the derivative with respect to the `y` axis you will get an expression representing the rate of change or slope in the `y` direction, while if you were instead take the derivative with respect to the `x` axis you would get an expression representing the slope in the `x` direction. The process to take that derivative is as simple as treating the other variable as if it were a constant like the `7`.

`(dz)/(dy)=7*x`

`(dz)/(dx)=7*y`

Answer 2

`(7xy)'=7xy' + 7y`

To get this we use the product rule:

`d((uv))/(dx)=u.(dv)/(dx)+v.(du)/(dx)`

So `d((7xy))/(dx) = 7x.dy/dx+ 7y.dx/dx`

Which we can write:

`(7xy)'=7xy'+7y`

or

`(7xy)'=7(xy'+y)`