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, extremaetc.
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¡¯s 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.
E) And here comes your Implicit Differentiation Calculator showing Step by Step solutions.
Calculus Made Easy for the TI89 is your best friend: