The slope of the tangent is the value of the derivative at that point

A function has a tangent at a point, and the slope of the tangent is the derivative of the function at that point

In the tangent slope of curve y = x ^ 3 + 3x ^ 2 + 6x-10, what is the tangent equation with the lowest slope?

Derivation = 3x ^ 2 + 6x + 6
With square = 3 [(x + 1) ^ 2 + 1], we can know that when x = - 1, the slope is the smallest = 3 in the tangent slope of curve y = x ^ 3 + 3x ^ 2 + 6x-10
Substituting x = 3 into y = x ^ 3 + 3x ^ 2 + 6x-10, there is y = 62
Let the tangent equation with the smallest slope be y = KX + B, and substitute k = 3, y = 62, x = 3 to solve B!

Among the tangents of curve y = X3 + 3x2 + 6x-1, the tangent equation with the smallest slope is () A. 3x-y-2=0 B. 3x+y+2=0 C. x+3y-2=0 D. x-3y+2=0

∵y=x3+3x2+6x-1，
∴y′=3x2+6x+6=3（x+1）2+3≥3．
When x = - 1, y ′ min = 3,
At this time, the slope is the smallest, that is, k = 3
When x = - 1, y = - 5,
This tangent passes through (- 1, - 5),
The tangent equation is y + 5 = 3 (x + 1),
That is 3x-y-2 = 0
So choose a

Curve y = x cubic + 3x quadratic + 6x-10, the tangent equation with the smallest slope is?

y'=3x ²+ 6x+6=3(x+1) ²+ three
Y 'is the smallest if the tangent slope is the smallest
At this time, x = - 1, y '= 3
y=0
So tangent point (1,0)
So 3x-y-3 = 0

Find the slope of tangent line of curve y = x ^ 2 + 3x + 1 at point P (1,5) and the equation of tangent line The correct answer is k = 5, y = 5x. Please explain in detail

y=f(x)=x^2+3x+1
f'(x)=2x+3
f'(1)=2+3=5
f(1)=1+3+1=5
The tangent slope of F (x) at x = 1 is 5, and the tangent passes through the point (1,5)
The tangent equation is Y-5 = 5 (x-1)
Sorted y = 5x

Is finding the first derivative the tangent slope of the curve? What about the left and right derivatives? Isn't there only one slope?

For continuous functions, if the left and right derivatives are equal, there is naturally only one slope, but for discontinuous functions with discontinuous point x = a, the left and right derivatives are not necessarily equal. The landlord thinks for himself that for a discontinuous function, his expressions on the left and right sides of x = a are not necessarily equal. Can you say that there is only one slope? At this time, we should judge whether it is differentiable at this point by whether the left and right derivatives are equal

The slope of the tangent is the value of the derivative at that point