Implicit Differentiation

Related Topics:

• Multivariable Calculus
• Logarithmic Differentiation

Conceptualisation

Imagine a (circle) function:
$S(x,y)=x^2+y^2$
Implicitly differentiating, you get:
$dS=2xdx+2ydy$
But what does that mean? Well a tiny difference in $S$ ($dS$) is equal to the above. (Note: It’s dependent on the tiny differences you moved $x$ and $y$ but also $x$ and $y$ themselves).

What if your formula started off as:
$S(x,y)=x^2+y^2=5^2$
If you differentiate, you get there’s no difference in the $S$ (I.E. $dS=2xdx+2ydy=0$).
$\therefore$ Your value of function $S$ won’t change.
$\therefore$ You can rearrange $dS$ to be the slope on the circle.