The relationship between the derivative equation and the tangent equation

You set a parabola, let's say y=3xx+2x+1, and take a little bit (1,6) on it

After (1,6) to make a tangent, you should be able to calculate this tangent, using the most commonly used discriminant method, so that δ=0 can be found.

y=8x-2 This is the tangent equation for the point (1,6).

And then there's the point:

You take the derivative of the tangent equation and get y=8, which means that the slope of the tangent is 8, right?

You derive the curve equation and get y=6x+2 to get the equation for a straight line. What does this mean?

This means that the slope of the curve (i.e., the parabola) varies with x. If you take x=1 and add the derivative of the curve y=6x+2, you do the math, you get 8, right?

This shows that when x = 1, the tangent slope of the parabola point is 8.

That is, the derivative of an equation that shows what the slope of the slope is when the curve takes different x values.

You can see it when you draw it.

y=3xx+2x+1, when x goes from - to +, its tangent slope is increasing all the time.

On the left side of the axis of symmetry, the slope is negative, on the axis of symmetry the slope is 0, and on the right side of the axis of symmetry, the slope is positive.

This is consistent with the derivative function of the parabola that we have obtained as y=6x+2.

y=f(x)

Derivative equation: y=f'(x)

Tangent equation: a, b) = tangent at (a, f(a)) point:

y = f'(a)(x-a) +f(a)

relation, except that the slope of the tangent equation at the point (a,f(a)) is the value f of the derivative equation at the point x=a'(a)

The tangent equation formula for the derivative: (y-b)=k(x-a).

Find the derivative of the function at (x0,y0) first, and the derivative value is the slope value of the tangent of the function at x0. After substituting the point coordinates (x0, y0), the tangent equation can be obtained by using the point oblique formula.

When the derivative value is 0, the tangent of the change point is y=y0; When the derivative does not exist, the tangent is x=x0; When it is not derivable at that point, there is no tangent. Derivative, also known as derivative value.

Tangent equation with p as tangent point: y-f(a)=f'(a)(x-a); If there is a tangent of the curve c through p, and the tangent point is q(b, f(b)), then the tangent is y-f(a)=f'(b)(x-a), also y-f(b)=f'(b)(x-b), and [f(b)-f(a)] b-a)=f'(b).

If a point is on a curve:

Let the curve equation be y=f(x), and a branch-carrying point on the curve is (a,f(a)).

Find the derivative of the curve equation and get f'(x), substituting a point to get f'(a), which is the tangent slope of the point (a, f(a)), which is obtained from the point oblique equation of the straight line. y-f(a)=f'(a)(x-a)

If a point is not on the curve:

Let the curve equation be y=f(x) and a point outside the curve is (a,b).

Find the derivative of the curve equation to obtain f'(x), let the tangent point be (x0,f(x0)), and substitute x0 into f'(x) to get the tangent slope f'(x0), from the point oblique equation of the straight line, the equation of the tangent line y-f(x0)=f'(x0)(x-x0), because (a,b) is on the tangent, substituting the obtained tangent equation, has: b-f(x0)=f'(x0)(a-x0), x0 is obtained, and the tangent equation obtained by substitution is obtained, that is, the tangent equation is obtained.

Tangent. The basics of problem failure.

a) Definitions related to tangents.

1. Definition of tangent line: take the point B near a point A of the curve, and make B continue to approach A along the curve. In this way, the limit position of the straight line ab is the tangent of the curve at point a.

1) This is the exact definition of the tangent, on the one hand, it can be qualitatively understood in the image as the straight line just touches the curve, on the other hand, it can also be understood as a dynamic process, so that the point B near the tangent point A is constantly approaching A, and when the distance from A is very small, observe whether the straight line AB is stable in a position.

2) To determine whether a straight line is a tangent of a curve, it is no longer possible to judge by the number of source empty ages of common points. For example, functions.

The tangent at (-1, -1) has two points in common with the curve.

3) In the definition, the point B is constantly approaching the point A in two directions, the point to the right of point A is approaching to the left, and the point on the left is approaching to the right, and this limit position can only become a tangent at point A if the limit position of the line AB is unique regardless of which direction it approaches. For a continuous function.

There is no guarantee that there will be tangents at every point.

For example, y=|x|At (0,0), the secant line is cut when the point to the left of x=0 is infinitely close to it.

The limit position of y=-x , and when the point to the right of x=0 approaches it infinitely, the limit position of the secant is y=x, and the limit position of the two different directions is not the same, so y=|x|There are no tangents at (0,0).

4) Since point b is constantly approaching a along the curve of the function, if f(x) has a tangent at a, then it must be defined (both left and right) at and near point a

3. From the geometric meaning of derivatives, several types of points without derivatives can be explained by combining numbers and shapes:

1) Boundary points of the function: The points to the left (or right) of such points are not in the defined domain.

, so that there is no secant line on one side, and there is no way to talk about the limit position. Therefore the tangent does not exist, and the derivative does not exist; Similarly, there are piecewise functions.

If it is not continuous, there is also no derivative for the boundary value at the break.

2) If the secant limit position of the known point is not the same as that of the point near the left and right, there is no tangent line, so there is no derivative. For example, in the previous example, y=|x|There is no derivative at (0,0). This kind of situation mostly occurs at the boundary of monotonic interval changes, and it is only necessary to select a point to approach the known point to observe whether the limit position is the same.

3) If there is a tangent at a known point, but the tangent is perpendicular to the x-axis, its slope does not exist, and the derivative does not exist at that point. For example:

Not directable at (0,0).

In summary: none of the points discussed in (1)-(3) have a derivative, while (1) and (2) have no tangent, and the points in (3) have tangents but no derivatives. It can be seen that if there is a derivative at a certain point, there must be a tangent, and if there is a tangent, there may not be a derivative.

y=f(x)

Derivative equation: y=f'(x)

Tangent equation: a, b) = (a, f(a)) on the point of the song cong cut cover line:

y = f'(a)(x-a) +f(a)

relation, except that the slope of the tangent equation at the point (a,f(a)) is the value f of the derivative equation at the point where the object Sakura x=a'(a)

The tangent equation formula for derivatives is as follows: the derived value is used as the slope k and then the original point (x0, y0) is used, and the tangent equation is (y-b)=k(x-a).

Method for finding tangent equations for derivatives.

Calculate the derivative f first'(x), the essence of the derivative is the slope of the curve, for example, there is a point ( on the function, and the derivative f of that point'(a)=c then the tangent slope of the point k=c, assuming that this tangent equation is y=mx+n, then brother burns m=k=c, and ac+n=b, so y=cx+b-ac

Formula: The derived value is used as the slope k and then the original point (x0, y0) is used, and the tangent equation is (y-b)=k(x-a).

The algorithm of derivatives

Subtraction law: envy type virtual (f(x)-g(x)).'f'(x)-g'(x)

Addition rule: (f(x)+g(x)).'f'(x)+g'(x)

Multiplication rent model: (f(x)g(x))).'f'(x)g(x)+f(x)g'(x)

Division rule: (g(x) f(x))).'g'(x)f(x)-f'(x)g(x))/f(x))^2