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In calculus, a derivative is the instantaneous rate of change of a function with respect to one of its variables.

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The tangent of the inclination angle α of a straight line is called the slope of this line.

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Acceleration is the essence of dynamism. It's not just about raw speed, but about the thrill of the change itself.

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Acceleration is the rate of change of velocity of an object with respect to time.

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Slope is a numerical measure that describes the steepness and direction of a line (or the tangent to a curve at a specific point).

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Slope can be positive or negative, and has magnitude.

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A value that can describe the instantaneous rate of change between variables x and y.

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The slope of a line is a measure of its steepness. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.

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The curve y=x2 does not have a constant slope.

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The slope indicates the direction and steepness of the line. 

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Tangent line of a curve

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A line which only touch a point of a curve and match the slope of the curve on this point.

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A line crossing acurve at two points is not the tangent line.

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Its slope is same as the curve on this point.

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the tangent line of y=x∧2 on (0，0) is y=0

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the process of getting the derivatives of a function.

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the process of getting the derivatives of a function.

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slope equals y1-y2/x1-x2 (x1,y1),(x2,y2)is on the line

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In geometry, a secant line is any line that intersects a curve in at least two points. 

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A straight line that intersects a curve at two or more distinct points.

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Let's take the function  f(x) = x^2 .

Points: Choose two points on the curve: (1, 1) and (3, 9).

Secant Line: The straight line connecting these two points is a secant line.

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Indicate the deepness of a curve at one point

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A number used to indicate the steepness of a curve at a particular point.

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It is defined to be the slope of the tangent line.

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Ice Bear

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The value a function approaches as the input gets closer to some point.

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The total distance of a trip (that's integration).

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Predicts behavior near a point, not necessarily at the point. Foundation for all of calculus.

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Calculating your exact speed at a specific moment on your speedometer.

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The Big Idea: It's All About the Tangent Line

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This special line is called the tangent line.

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Of course! Let's break down the concept of the "slope of a curve."

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Unlike a straight line, a curve has a slope that changes from point to point. Therefore, when we talk about the "slope of a curve" at a given point, we are really talking about the slope of the straight line that just "touches" the curve at that point.

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The limit of the average rate of change as the time interval approaches zero.

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Rate of change of a single point in time.
Defined as the derivative of a function.

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The rate of a chemical reaction at a precise time.

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A tangent is a straight line that touches a curve at a single point without crossing it.

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It only have one crossover point with the line.

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y=x*x,when x=1,its tangent line of a curve is y=2*x-1

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the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivativeat.The derivatives occurs in many applications across many disciplines. These applications include velocity and acceleration in physics, marginal profit functions in business, and growth rates in biology.

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An expression of the rate of change at one point

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The slope of a line is a number that describes both the direction and the steepness of the line. And the formula is: m = (y₂ - y₁) / (x₂ - x₁) = Δy / Δx = Rise / Run

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Non-examples are relationships that are not straight lines and therefore do not have a single, constant slope.    Examples: A parabola (y = x²), a circle (x² + y² = 9), or an exponential curve (y = 2ˣ).

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It have direction including positive slope,negatived slope,zero slope, undefined slope 

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Equation Example: y = 2x + 1 For every 1 unit you move to the right (run), you move up 2 units (rise).So its slope of the line is 2.

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Phoenix

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The derivative of a function 
f(x)
 at a point 
x
 is defined as the limit of the difference quotient as the interval 
h
 approaches 0:

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 constant value like 
7
 (while the derivative of a constant function 
f(x)=7
 is 
0
, the number 
7
 itself is not a derivative describing a varying rate of change).

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Represents the rate of change of a function with respect to an independent variable.

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The change in position of an object divided by the total time taken for that change 

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Saying a runner’s average velocity was 5m/s without specifying the direction.

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· The slope of the secant line between two points on a position-time graph.

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A car travels 50km North in 1 hour.Its average velocity is 50km/h,North.

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The ratio of the vertical change to the horizontal change between any two points.

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Measures the steepness and direction of a line.

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The equation y = 3x - 5 has a slope of 3.

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measure the steepness and direction of a line

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y＝-9x＋2，in this equation，-9 is the slope

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Differentiation‌ is the process of finding the ‌derivative‌ of a function, which measures how a function's output changes with respect to its input.

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A general call of the methods using calculus to solve math problems.It always needs a mathematical model to calculate.

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When we need to measure a container’s volume according to its given data,calculus will be a great helper.

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velocity

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a measure of how fast an object is moving. It is the rate at which an object covers distance.

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At its heart, the slope of a line is a number that describes the steepness, direction, and incline of a line. It is a measure of how much a line rises or falls as you move from left to right along it.

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At its heart, the slope of a line is a number that describes the steepness, direction, and incline of a line. It is a measure of how much a line rises or falls as you move from left to right along it.

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Think of it as the "slant" of the line. We encounter slope everywhere in the real world:

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easy

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describe the total rate of change in a period,but not the true rate

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DD🐟'rate of change of velocity

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The rate of change of velocity.It measures how quickly the swimming speed of DD🐟 is changing.

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DD🐟 takes a deep breath to speed up!!! From 10m/s to 100m/s in 9 seconds.The acceleration is 10m/s².

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it can present the rate change at one point and it can be presented by the eaquation of f'(x)

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When the f(x)=Sinx the derviate can be x

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The slope of a line is basically how steep it is.Think of climbing a hill - a big slope means it's really steep. We calculate it by how much the line goes up or down divided by how far it goes across. If a line slopes upwards, it's positive; if it slopes down, it's negative. This idea is everywhere in real life - like when you see how quickly prices are rising, or how fast a car is going. It's one of those math concepts that's actually really useful to know!

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The slope of a line is basically how steep it is.Think of climbing a hill - a big slope means it's really steep. We calculate it by how much the line goes up or down divided by how far it goes across. If a line slopes upwards, it's positive; if it slopes down, it's negative. This idea is everywhere in real life - like when you see how quickly prices are rising, or how fast a car is going. It's one of those math concepts that's actually really useful to know!

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a line cross at least one point in the function

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y=kx+b two points on the function(x1,y1)and (x2,y2)are involved in this line

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The degree of inclination of a certain line

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y=x*x we cannot say it has a exactly slopo

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It describes the steepness and direction of a line.

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It can be many numbers.And it is negative or positive.

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The slope  of a line is a number that describes both the direction and the steepness of the line.
The most common way to calculate slope is "rise over run." If you have two points on a line, Point 1: (x₁, y₁) and Point 2: (x₂, y₂), the slope is calculated as:

m = {y_2 - y_1}/{x_2 - x_1}

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Such as(3,9)(5,10)two points,10-9=1isn't the slope of a line.

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Let's find the slope of the line that passes through the points (2, 3)and (5, 11).
Step 1: Identify the coordinates.
   Point 1 (x₁, y₁) = (2, 3)
   Point 2 (x₂, y₂) = (5, 11)

Step 2: Apply the slope formula.

m = {y_2 - y_1}/{x_2 - x_1}


Step 3: Substitute the values.

m = {11 - 3}/{5 - 2}


Step 4: Calculate.
m = 8/3
The slope of the line is m = 8/3

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it describes the function's derivative in that point

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Differentiation is the process or result of differentiation, with the core idea of infinite division, describing the linear approximation of a function in small variations

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This linear approximation can describe the approximate curve of a function in a local region, especially geometrically manifested as the approximation of a tangent to the curve.

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A way using calculus tocalculate some mathematical model and find the derivatives of a function.