Implicit Differentiation Technique, Formula & Examples

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  • 0:04 Implicit…
  • 0:54 Explicit vs Implicit
  • 3:55 Implicit…
  • 6:08 Some More Examples
  • 8:03 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Differentiation techniques include implicit differentiation. In this lesson we define the method of implicit differentiation and demonstrate this technique with numerous examples.

Implicit Differentiation Technique

Let's say that our friend Gary is letting us know that he's going to be late for our 4:15 PM supper meeting, but all he says is that there's still lots of homework that he has to do. When asked to be clearer, Gary explicitly shares his schedule: at 4 PM he studies math, at 4:10 he takes a break, and at 4:20 he starts the homework. By explicitly stating his schedule, we know for sure Gary has no plans to be at the restaurant by 4:15.

In math differentiation problems, we often hear these same words: explicit and implicit. In this lesson we explain the implicit differentiation technique by comparing it to explicit differentiation. We then complete several examples using implicit differentiation. This might help Gary make it to the restaurant.

Explicit vs. Implicit

Given the equation y2 = x - 1, let's find the tangent line. We know taking the derivative of y with respect to x gives us the slope of the tangent line. If we first take the square root of both sides of this equation, we have:


y_explicit


This is the explicit form for the equation because y is isolated on the left-hand side and x appears only on the right-hand side.

Differentiating and solving for y', we get the equation below:


y_explicit_differentiation


The first line says y' is the same as dy / dx is the same as D y. The prime notation, the d over dx notation, and the capital D notation are three ways to say the same thing: take the derivative with respect to x. It will be convenient to mix these notations.

The second line is the derivative of a square root, and the third line is the final answer where we are careful to avoid values for x undefined for y' (x = 0 leads to dividing by zero).

Differentiating an explicit form equation is called explicit differentiation.

Let's get the y' answer without first solving for y. Using the capital D notation for derivative, take the derivative of both sides of the equation:

D(y 2) = D(x -1) which gives:

2y y ' = D(x) - D(1) = 1 - 0, having used the chain rule for D(y 2). The derivative of a difference is the difference of the derivatives. Thus, D(x - 1) is the same as D(x) - D(1). The derivative of x is 1, and the derivative of a constant (the 1) is zero.

2y y' = 1, or y' = 1/(2 y). See how nicely the y' appears letting us easily solve for y'?

The chain rule is nicely expressed using the d dx notation:


chain_rule


See how the y' replaced the dy dx? Equivalent and more compact!

Substituting for y = square root(x - 1) we get:


y_implicit_differentiation


The results are the same, but the second approach is very powerful when it's impossible or inconvenient to explicitly solve for y. This second approach is called implicit differentiation, which is essentially accomplished using the chain rule.

And one rule of effective study is to take breaks, including eating. Will Gary continue studying and miss supper?

Implicit Differentiation Example

Let's complicate the previous equation by mixing in more x and y terms:

(x - y)2 = x + 8y - 1. A plot of this curve looks like the image below with this equation:


A plot of the curve
the_curve


Find the tangent line at the point x = 5, y = 1.25:


The point for the tangent line
the_point


Differentiation gives the slope of the tangent line, but instead of trying to solve for y to do an explicit differentiation, we'll differentiate implicitly. First, take the derivative of both sides of the equation:

D (x - y)2 = D (x + 8y - 1)

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