Implicit differentiation
Bit hard to understand it in the first place.
What is Implicit & Explicit Function
Refer to video by Krista King: What is implicit differentiation?
Explicit function
: it's the normal function we've seen a lot before, which's in the form ofy = x....
Implicit function
: it't NOT YET in the general form of a function and not easily separated, likex² + y² = 1
So knowing how to differentiate an implicit function
is quite helpful when we're dealing with those NOT EASILY SEPARATED functions.
How to Differentiate Implicit function
Refer to video: Use implicit differentiation to find the second derivative of y (y’’) (KristaKingMath)
Refer to video by The Organic Chemistry Tutor: Implicit Differentiation Explained — Product Rule, Quotient & Chain Rule — Calculus
Refer to Symbolab: Implicit Derivative Calculator
Assume you are to differentiate Y
WITH RESPECT to X
, written as dy/dx
:
- Differentiate terms with
X
as normal - Differentiate terms with
Y
as the same toX
, BUT multiply by(dy/dx)
- Differentiate terms MIXED with
X & Y
by usingProduct Rule
, then differentiate each term.
How to differentiate Y with respect to X
How to differentiate term MIXED with both X & Y
Example
Solve:
Refer to Symbolab: Implicit Derivative Calculator
- Treat
y
asy(x)
- Apply the Sum Rule:
- Apply the normal rules to
X term
, and - Apply the Product Rule to the
Mixed term
, and - Apply the Chain Rule to the
Y term
:
- Operate the equation and solve for
dy/dx
, and get:
Example
Solve:
- First thing we need to find the RIGHT equation of Chain rule. Since it’s asking us to find
dy/dt
, so we will re-write it to this one to form an equation:
- Then since we’ve given the
dx/dt = -3
, we only need to find out thedy/dx
to get the result. - We’ve got an equation of
x & y
, regardless whom it's respecting to. So we can do eitherImplicit or Explicit differentiation
to the equationy²=7x+1
, with respect toy
:
- Use the implicit differentiation method, we got the
dy/dx = 7/2y
- And since
y=6
, so7/2y = 7/12
- Back to the Chain Rule equation, we get
dy/dt = 7/12 · (-3) = -7/4 = -1.75
Example
Solve:
- Remind you that, in this problem, it’s NOT respecting to
x
anymore, so you need to change mind before getting confused. - First thing we need to find the RIGHT equation of Chain rule. Since it’s asking us to find
dx/dt
, so we will re-write it to this one to form an equation:
- Then since we’ve given the
dy/dt = -0.5
, we only need to find out thedx/dy
to get the result. - We’ve got an equation of
x & y
, regardless whom it's respecting to. It seems easier to differentiate explicitly:
- Then we use
d/dx
to differentiate the equation to get:dx/dy = y⁻² = (0.2)⁻² = 25
- Back to the Chain Rule equation, we get
dx/dt = dx/dy · dy/dt = 25 * (-0.5) = -12.5
.
Example
Solve (Same with above examples):
- Form an equation:
dx/dt
has been given equals to5
, so just to find outdy/dx
:
- And get:
- Now let’s see what is
sin(x)
equal to:
- All done.
Vertical & Horizontal Tangents of Implicit Equations
► Jump over to Khan academy for practice.
Example
Solve:
- Plug in
y = 0
into the equation and get thatx = -6
, which is the answer.
Example
Solve:
- To have a
Vertical Tangent
, we have to let the derivative becomeUndefined
, - which in this case is to let the denominator equal to zero:
- Solve this equation out we get that
x = 3y²
, which means this relationship is true at the point of vertical tangent line. - Plug that back to the original function to get
y = -1
, which means the vertical tangent goes through this point. - Substitute y back and get
x = 3
- The answer is
(3, -1)
.