Implicit differentiation

Bit hard to understand it in the first place.

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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 of y = x....

• Implicit function: it't NOT YET in the general form of a function and not easily separated, like x² + 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 to X, BUT multiply by (dy/dx)

• Differentiate terms MIXED with X & Y by using Product Rule, then differentiate each term.

How to differentiate Y with respect to X

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How to differentiate term MIXED with both X & Y

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Example

Solve: Refer to Symbolab: Implicit Derivative Calculator

• Treat y as y(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 the dy/dx to get the result.

• We've got an equation of x & y, regardless whom it's respecting to. So we can do either Implicit or Explicit differentiation to the equation y²=7x+1, with respect to y:

  
• Use the implicit differentiation method, we got the dy/dx = 7/2y

• And since y=6, so 7/2y = 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 the dx/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 to 5, so just to find out dy/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 that x = -6, which is the answer.

Example

Solve:

• To have a Vertical Tangent, we have to let the derivative become Undefined,

• 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).