STEP BY STEP Implicit Differentiation with examples-
Learn how to do it in either 4 Steps or in just 1 Step.
A) You know how to find the derivatives
of explicitly defined functions such as y=x^2 ,
y=sin(x) , y=1/x, etc .
What if you are asked to find the
derivative of x*y=1 ? This is an Implicitly
defined function (typically a relation) as y is not alone on the left side of
the equation.
Well, one way would be to rewrite it as
an explicit function by dividing both sides by x to get y=1/x and we know its
derivative dy/dx = -1/x^2 using quotient rule.
Well, this Houdini "trick"
does not always work. You couldn't use it for x^2*y+y^3*x=6. So how can we find
its derivative?
First, we have to recall and use the
fact that y as the dependent variable depends on x, so we could rewrite the
equation as x^2*y(x)+(y(x))^3*x=6 .
This equation format convinces us i.e.
to use chain rule for (y(x))^3 yielding 3*(y(x))^2*(dy/dx) where dy/dx=y'(x) is the
derivative of y with respect to x.
Altogether, when applying product rule
(f*g)' = f'*g + f*g' we get : 2x*y(x)+x^2*(dy/dx) + 3*(y(x))^2*(dy/dx)*x+(y(x))^3=0 (This
is STEP 1)
Since we are after dy/dx , we have to factor it : (dy/dx)*[x^2
+ 3*(y(x))^2] + 2x*y(x) + (y(x))^3=0 (This is STEP 2)
And we have to subtract the non dy/dx to other side: (dy/dx)*[x^2
+ 3*(y(x))^2] =
- 2x*y(x) - (y(x))^3 (This is STEP 3)
Lastly, we just have to divide by the
non dy/dx on the left side: dy/dx = (- 2x*y(x) - (y(x))^3
) / [x^2 + 3*(y(x))^2] (This is STEP 4)
Congrats, you just mastered Implicit
Differentiation Step by Step (in 4 Steps to be precise) !!!!!
Lets
revert back to y instead of y(x) : dy/dx
= (-2x*y-y^3)/(x^2 + 3*y^2) to make our final derivative look less complicated.
VOILA!!!
Remember that this derivative evaluated
at (x,y)-points gives the
slope at those points and we can draw conclusion about in/decreasing, extrema etc.
B) Lets go back to x*y=1 . Try to find its derivative using the 4 Implicit Differentiation
Steps outlined above and compare it to the derivative found earlier.
Continue reading after performing the STEP BY STEP Implicit
Differentiation!
STEP 1 : 1*y + x*dy/dx =
0 by Product Rule.
STEP 2 + 3: x*dy/dx =
-y no factoring here and subtract the
non dy/dx.
STEP 4 : dy/dx = -y/x DONE. Now how does -y/x match -1/x^2 ? Algebra helps here: Since y=1/x :
-y/x = -(1/x)/x = -1/x^2 and everything that starts well ends well ;-)
C) IMPLICIT DIFFERENTATION in only 1 STEPS ? Is this Possible???? Yes it is
....
To accomplish this, we have to subtract
whatever is on the right side over the left side. In our example above, we get
x^2*y+y^3*x-6=0, lets name this F=0 . Easy enough.
Now we use: dy/dx = -Fx/Fy that is our little
formula.
Fx
= is the derivative of the left side where x is our variable and the y's are
treated as constants (Think of y as y=10) . In our
example, Fx= 2*x*y+y^3
Similarly, Fy = is the derivative of the left side where y is
our variable and the x's are treated as constants. (Think of x as x=10) . Fy =
x^2 + 3y^2*x
After this prep work ,
we find dy/dx = -Fx/Fy = -(2*x*y+y^3)/(x^2 + 3*y^2*x)
Now, how cool is this
?? Implicit Differentiation in only 1 STEP plus a bit of prep work.
Can you do Implicit Differentiation in 1
step for x*y=1 ? Or x*y -1 = 0
Here is how: Fx
= y , Fy = x . Thus, dy/dx
= -Fx/Fy = -y/x Done.
D) SUMMARY: Make an implicitly defined
function explicit, if possible, by solving for y and differentiate it as such.
If impossible, do Implicit
Differentiation in 4 Steps as outlined in B) .
And if you are not afraid of finding the
partial derivatives Fx and Fy then do Implicit Differentiation in just 1 step. Though,
your teacher may not like it as the typical 4 steps in B) are not included and
your final solution looks if copied.