Find an equation of the tangent line to the given curve at...

Key Concepts

Tangent Line

A tangent line to a curve at a specific point is the straight line that just touches the curve at that point without crossing it. Its slope is equal to the derivative of the function at that point, making it a key concept for understanding local linear approximations.

Point-Slope Form

The point-slope form of a line is a way to express the equation of a straight line given a point on the line and its slope. It is expressed as y - y? = m(x - x?), where m is the slope and (x?, y?) is the known point.

Derivative

The derivative of a function measures the instantaneous rate of change or the slope of the function at any given point. It is a fundamental concept in calculus used to analyze and understand the behavior of functions, particularly how they change with respect to their input variables.

Quotient Rule

The quotient rule is a differentiation technique used when dealing with functions represented as the ratio of two differentiable functions. It provides a systematic method for finding the derivative of a quotient, ensuring that both the numerator and denominator are appropriately handled.

Transcript

00:01 All right, so in this problem, we've got a function here, y equals x squared over 1 plus x, okay, in black.

00:08 And then we've got a point here, one, one half.

00:15 All right.

00:16 And our goal is to find the equation of the tangent line.

00:20 Okay, so equation of the tangent line.

00:32 So if we want to find the equation of the tangent line, well, we've got a point.

00:39 Point, okay, that we want the tangent line to touch.

00:43 All right, so we've got a point on the line.

00:45 We need the slope of the line.

00:47 All right.

00:48 And when we talk about derivatives, all right, derivative is the slope.

00:54 Okay, so if we find the derivative and then we plug in the x value of our point, we should get out the slope at that point.

01:06 That's what the derivative is.

01:07 It's keeping track at the slopes all across.

01:11 As we move across x values okay so first things first let's take the derivative of this x squared over one plus x all right well we've got a fraction right we've got an x squared on the top one plus x on the bottom so we're going to use quotient rule all right and that is in green here so our f needs to be the thing on the top x squared our g needs to be the thing on the bottom 1 plus x and in green there we can see that we're going to need f prime and g prime f prime all right the derivative of x squared will we bring the two down x to the first right we lower the power by one for g prime 1 is just a constant so the derivative it is zero plus the derivative of x well the derivative of x is just 1 right it's x the first bring the 1 down into the invisible coefficient of 1 already there and you get x to zero is another one a whole bunch of ones multiplied okay um so here we are we are ready to go find this derivative okay so let me just go and label it derivative keep track because now we've got multiple things going on here first things first g okay that's one plus x all right times f prime that's 2x all right the next step is minus f, that is x squared, times g prime, that is one, all over g squared.

03:09 Okay, so one plus x squared.

03:16 Now, we can simplify this.

03:19 We can do all the distribution and stuff.

03:22 But for this problem, what we care about is the slope...

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