Use implicit differentiation to find an equation of the...

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations in which the variables are intermixed rather than isolated. Instead of solving for one variable explicitly in terms of the other, both sides of the equation are differentiated with respect to the independent variable, treating the dependent variable as a function of the independent variable. This method is especially useful for finding derivatives of curves defined implicitly.

Chain Rule

The chain rule is a fundamental calculus tool used to differentiate composite functions. When applying implicit differentiation, the chain rule allows us to handle terms involving the dependent variable, as these terms are understood to be functions of the independent variable. This rule ensures that the derivative of such terms properly reflects the dependence on the inner function.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve’s instantaneous rate of change. Finding the tangent line involves first computing the derivative to obtain the slope at the point of tangency, and then using this slope along with the point to write the line’s equation.

Point-Slope Form

The point-slope form of a linear equation is a way to express the equation of a line given a point on the line and its slope. It is generally written as y - y? = m(x - x?), where m represents the slope and (x?, y?) is the given point. This form is particularly useful for writing the equation of the tangent line once the slope has been determined through differentiation.

Transcript

00:02 In this problem, we want to use implicit differentiation to find an equation on the tangent line to the curve at the given point.

00:11 And so when we're using implicit differentiation, we want to take the derivative of the equation in terms of x.

00:22 And so if we work term by term, then the derivative of x squared would be 2x.

00:32 And when we have two variables involved, so we have both an x and a y, we're going to have to use the product rule.

00:42 And with that, you're going to take and find first the derivative of the x, and the derivative of x is one.

00:55 And so find the derivative of x, but we're going to keep everything else the same.

01:00 And so the derivative of x is one, and so we keep the two y.

01:04 And then we're going to add to that now we're going to take the derivative of y with respect to x.

01:15 And so the derivative of y is going to end up being one.

01:23 We keep the two and we keep the x.

01:26 But when we take that derivative, because we're doing this with respect to x, we need to keep essentially a dy dx on there.

01:38 And d -y -d -x can be written as y prime.

01:42 And the idea is that since y is a function of x, this part of it would be the chain rule, you would take the derivative of the y, which would be kind of your outside function in the chain rule, and then you have to take it times the derivative of the inside, but we don't know what the inside is, and so we just say y -prime, so that we could essentially end up taking the derivative of that if we knew what y was.

02:09 And so then we're going to take the derivative of 4y squared, and the derivative of 4y squared would be 8y, but again, we need to keep that y prime on the end of there, that d, y, d, x.

02:25 And then the derivative of 12 is 0.

02:28 And once we get to this point, then you just need to essentially solve 4y prime.

02:36 And so in order to do that, we're going to keep the y primes on one side of the equation and put the x, the ones without the y primes on the other side.

02:46 And so we would do 2x y prime plus 8 y prime, and subtract your 2x over and subtract your 2y over...

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