Use implicit differentiation to find d y / d x. (3 x y+7)^2=6 y

Name: Problem 6, Chapter 3: Thomas Calculus
Uploaded: 2019-05-10T23:39:51-08:00
Duration: 4 min 44 s
Channel: Karl Schaefer
Description: Use implicit differentiation to find d y / d x. (3 x y+7)^2=6 y

Use implicit differentiation to find d y / d x. (3 x...

Key Concepts

Implicit Differentiation

Implicit differentiation is a method used when dealing with equations where y is defined implicitly as a function of x. Instead of solving for y explicitly, you differentiate both sides of the equation with respect to x, treating y as a function of x and applying differentiation rules accordingly.

Chain Rule

The chain rule is a fundamental differentiation rule used to differentiate composite functions. When differentiating an expression that involves a function inside another function, such as a power function of an algebraic expression containing both x and y, the chain rule helps by taking the derivative of the outer function and then multiplying by the derivative of the inner function.

Product Rule

The product rule is used when differentiating a product of two functions. In cases where an expression involves the product of x and y, the product rule provides a way to obtain the derivative by differentiating each function separately and adding the results together in a specific manner.

Transcript

00:01 And this problem, we're asked to use implicit differentiation to find d, y, d, d, y, x, where we have the implicit equation 3xy plus 7 all squared equals 6y.

00:11 So to do implicit differentiation, we're just going to take the derivative of the entire expression.

00:16 So we're going to take the derivative of both sides.

00:19 So on the left -hand side, that's the derivative of 3xy plus 7 squared equals the derivative of 6y with respect to x.

00:32 Now, this left -hand side is kind of a monster here.

00:35 There's a lot of stuff going on the left -hand side.

00:37 So let's just focus on the outer bit of this.

00:40 So i'm just going to kind of squint my eyes and forget about the middle.

00:45 So the left -hand side looks like blah squared.

00:47 How do we take the derivative of something squared? well, we need to use the chain rule, right? so the chain rule is exactly for situations like this, where we have an inner function, which i've called blah, and an outer function, which is squaring.

01:05 So what's going to happen with the chain rule? well, i should take the derivative of the outer function and plug in the inner function, so two times blah to the power of one, times the derivative of the inside.

01:20 So this is exactly what the chain rule tells us.

01:22 We take the derivative of the outside, we plug in what was on the inside, and then we multiply by the derivative of the inside.

01:29 So now let's substitute back what our original expression was.

01:32 So 2 times 3xy plus 7 times the derivative of 3xy plus 7.

01:43 So that's, this part was here.

01:47 So that's 2 times 3xy plus 7.

01:50 Now we need to differentiate 3xy plus 7...

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