I'm learning calculus II. I recently wondered what if I had two unknown variables in an function, and wanted to take an derivative.
Let's say there is a function $f(x,y)=2x^3+7y^2$
How would I calculate $\frac{df}{dx}$ ? What about $\frac{df}{dy}$?
$\endgroup$2 Answers
$\begingroup$When you're differentiating with respect to $x$ , $y$ is constant. So just treat $y$ as constant (given that $y$ is not a function of $x$)
So
$$\frac d{dx} \left( 2x^3+7y^2\right)=6x^2+0=6x^2$$
If $y$ is a function of $x$ then,
$$\frac d{dx} \left( 2x^3+7y^2\right)=6x^2+ 7.2y.\frac {dy}{dx} $$
The other part, if $x$ is not a function of $y$
$$\frac d{dy} \left( 2x^3+7y^2\right)=0+14y=14y$$
if $x$ is a function of $y$ then
$$\frac d{dy} \left( 2x^3+7y^2\right)=2.3x^2.\frac {dx}{dy}+14y$$
$\endgroup$ $\begingroup$$$\frac d{dx} \left( 2x^3+7y^2\right)=6x^2+0=6x^2$$
$$\frac d{dy} \left( 2x^3+7y^2\right)=0+14y=14y$$
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