f(x)=x^3 ; f'(x)=3x^2 ----- so how is 3x^2 the slope of tangent line?

Question

I'm a little confused. since the derivative of a function is equal to the slope of the tangent line, then what it is if is not a line? for example the derivative of x^3 is 3x^2. which is a parabola. so what does this signify?

Answer 1

Look at the equation x^3 which is an odd function that passes through the origin. If x=0, then f'(x)=3(0)^2=0 which is correct. So as another posted stated, for any value of x, 3x^2 is the slope of the tangent line at that point. So for x=-1, the slope would be 3(-1)^2=3. Don't look at 3x^2 as being a constant value, nor look at it as the equation of the tangent line. It is merely just the slope. Hope that clears it up.

Answer 2

The derivative of a function at a point is the slope of the tangent line _at that point_. This is always a single number. The derivative of a function f(x), when stated without qualification, is the function which, for every value of x, gives the derivative of f(x) at that point. Note that this is merely a formula giving you the slopes of many tangent lines, and is not itself a tangent line. Thus, the derivative function does not have to be a line.

Answer 3

To find out the slope of the tangent line you would plug in the value of x of the point in question. For example, when x=1, f(1)=1^3=1 and to find out the slope of the curve at that point plug x=1 into the equation f`(1)=3*1^2=3.

Answer 4

This means the slope of the tangent line is different at each value of x.

if you would like to find the slope of the line tangent to the equation f(x)=x^3 at x=2, then you would put 2 into the equation of f'(x): f'(2)=3(2)^2=12 to find the slope of the tangent line at that point is 12.

Answer 5

For all x, 3x^2 is the slope of f
Example: if you consider the point (1,1), (Remember that f(1) = 1^3 = 1) then you have this slope: f'(1) = 3*1^2 = 3*1 = 3

So, the tangent line is calculated using this formula:

y-y0 = m (x-x0)

m is f'(1) in this example, and x0= y0= 1

Just substitute and you will find the equation of a straight line


Ana

Answer 6

f (x) = 5 - 3 x ² f ` (x) = - 6x f ` (-1) = 6 = m is the slope

Answer 7

you'd better understand the meaning of derivative, think about the definition of derivative, the basic rule for you to work out the derivative, I remembered that it will use limit.