# Using derivatives to find equation for the graph

• donjt81
In summary, to find the equation of a graph using derivatives or anti-derivatives, the first step is to integrate the given differential equation. After finding the two unknown constants, solve for them using the given point and slope. Finally, substitute the values of the constants into the integrated equation to find the equation of the graph.
donjt81
Hi I have this problem where I have to find the equation of the graph using derivatives or anti-derivatives... I'm not sure... I really need some help on this...

Find the equation for the graph that passes through the point (-2,3) with the slope 1 given that d^2y/dx^2 = 6x/5

can someone point me in the right direction on how to start the solution.

donjt81 said:
Hi I have this problem where I have to find the equation of the graph using derivatives or anti-derivatives... I'm not sure... I really need some help on this...

Find the equation for the graph that passes through the point (-2,3) with the slope 1 given that d^2y/dx^2 = 6x/5

can someone point me in the right direction on how to start the solution.

This is a second order differential equation which can be solved easily by direct integration. After integrating, you'll have two unknown constants, which you will calculate with the help of the given slope and point.

Integrate
$$\frac{d^2y}{dx^2}= \frac{6}{5} x$$
to find dy/dx (plus a constant).
Integrate that to to find y (plus another constant).
Use the information given to determine what the two constants must be.

ok so y' = 3x^2/5 + C
y = x^3/5 + Cx + D

and also we have the point (-2,3) and slope 1
if we substitute the values for x and y here we get

3 = -8/5 -2C + D

?? how do we get the values of C & D... I'm stuck...

donjt81 said:
ok so y' = 3x^2/5 + C
y = x^3/5 + Cx + D

and also we have the point (-2,3) and slope 1
if we substitute the values for x and y here we get

3 = -8/5 -2C + D

?? how do we get the values of C & D... I'm stuck...
"also we have the point (-2,3) and slope 1" so you know that
y(-2)= 3 and y'(-2)= 1.

You have correctly put those values into the equation for y(x) to get one equation but you haven't used the fact that the slope there is 1. Now put x= -2, y'(-2)= 1 into the equation for y'(x) to get a second equation. It's easy to solve two linear equations for the two unknown values C and D.

ohhh ok so...

y' = 3x^2/5 + C
1 = 3(-2)^2/5 + C
C = -1.4

y = x^3/5 + Cx + D
3 = -2^3/5 - 1.4 * -2 + D
D = 1.8

So the eqn of the graph is
y = x^3/5 - 1.4x + 1.8

is that correct?

Check it yourself. If x= 1 what is y(x)? If x= 1, what is y'(1)? Does the second derivative satisfy the equation you were given?

## 1. What are derivatives and how are they used to find equations for a graph?

Derivatives are mathematical tools used to analyze the rate of change of a function. They are used to find the slope of a curve at a specific point, which can then be used to create an equation for the graph.

## 2. Can derivatives be used to find the equation for any type of graph?

Yes, derivatives can be used to find the equation for any type of graph as long as the graph is continuous and differentiable. This means that the curve must be smooth and have no sharp corners or breaks.

## 3. How do you use derivatives to find the equation of a line?

To find the equation of a line using derivatives, you need to find the slope of the line at a specific point. This can be done by taking the derivative of the function at that point. The equation of the line can then be written in the form y = mx + b, where m is the slope and b is the y-intercept.

## 4. Are there any limitations to using derivatives to find an equation for a graph?

Yes, there are limitations to using derivatives to find an equation for a graph. As mentioned earlier, the graph must be continuous and differentiable. Additionally, derivatives can only find the equation for a single point on the graph, so multiple points must be used to create a more accurate equation.

## 5. Can derivatives be used to find the equation for nonlinear graphs?

Yes, derivatives can be used to find the equation for nonlinear graphs as long as the graph is differentiable. However, the process may be more complex as the slope of a nonlinear curve can change at different points, so multiple derivatives may need to be taken to find the equation.

Replies
4
Views
1K
Replies
4
Views
687
Replies
35
Views
3K
Replies
24
Views
2K
Replies
19
Views
5K
Replies
3
Views
512
Replies
6
Views
2K
Replies
22
Views
2K
Replies
8
Views
3K
Replies
15
Views
2K