# What is the derivative of 7xy?

Dec 18, 2014

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

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.

$z = 7 \cdot x \cdot y$

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 $7$.

$\frac{\mathrm{dz}}{\mathrm{dy}} = 7 \cdot x$

$\frac{\mathrm{dz}}{\mathrm{dx}} = 7 \cdot y$

Dec 18, 2014

$\left(7 x y\right) ' = 7 x y ' + 7 y$

To get this we use the product rule:

$d \frac{\left(u v\right)}{\mathrm{dx}} = u . \frac{\mathrm{dv}}{\mathrm{dx}} + v . \frac{\mathrm{du}}{\mathrm{dx}}$

So $d \frac{\left(7 x y\right)}{\mathrm{dx}} = 7 x . \frac{\mathrm{dy}}{\mathrm{dx}} + 7 y . \frac{\mathrm{dx}}{\mathrm{dx}}$

Which we can write:

$\left(7 x y\right) ' = 7 x y ' + 7 y$

or

$\left(7 x y\right) ' = 7 \left(x y ' + y\right)$