Differentiation with Graphs

Gradients of Lines

The following graph shows the line y = 2x + 1

We know to find the gradient we can find 2 coordinates on the line and calculate \dfrac{\text{change in}  y}{\text{change in}  x}:

Using (0,1) and (3, 7):

\dfrac{7-1}{3-0} = 2

So, the gradient, or rate of change, of this line is 2.

 

We could also calculate this by differentiating the equation of the line:

y = 2x + 1\\

\dfrac{\text{d}y}{\text{d}x}=1\times 2x^{1-1} + 0 = 2

Gradients of Curves

We can also use differentiation to find the gradients of curves at given points.

The following graph shows the curve y = x^2 + 2

Curves have many gradients, depending on which point you are looking at.

For example:

Find the gradient of the curve y = x^2 + 2 at the point (2,6).

Differentiate:

\dfrac{\text{d}y}{\text{d}x} = 2x

Sub in the value of x from (2,6):

\dfrac{\text{d}y}{\text{d}x} = 2(2) = 4

Therefore, the gradient at the point (2,6) is 4.

If we were asked the same question but with the point (4, 18), we would substitute x=4 into the derivative:

\dfrac{\text{d}y}{\text{d}x} = 2(4) = 8

Stationary Points

A stationary point on a curve occurs when the gradient is 0.

The curve below has 2 stationary points, one that is a maximum point, and one that is a minimum point. 

The gradient at both the maximum and minimum point is 0.

 

Finding the Coordinates of Stationary points:

For example, if a curve had the equation y = 2x^2 + 4, we can use differentiation to find the coordinates of the stationary point(s).

\dfrac{\text{d}y}{\text{d}x} = 4x

 

At a stationary point, we know the gradient, 4x is 0:

4x = 0\\

 

We solve this equation:

x = 0

 

Therefore, the stationary point occurs when x=0. To find the full coordinate, we can sub this x coordinate back into the curve’s equation:

y = 2(0^2) + 4 = 4

 

Coordinate: (0,4)