Derivative, gradient and rate of change

Watch this to appreciate the connection between the derivative, rate of change and gradient of a curve for Higher Maths (and to look at the mathematics of change)

Things change.

Traffic flow changes, populations change, weather changes, the amount of money in an investment changes, speed changes when you accelerate.

Often we want to find out how things are changing – quickly or slowly.

The rates at which things change gives us information about how to respond to them. To understand why and how and how quickly things change, gives us the ability to develop insight and adjust to the world around us.

In this episode we are going to look at derivative as a rate of change or as a gradient.

This is thinkfour.

You mustn’t just think of a derivative as an expression you get when you differentiate a function.

It has meaning when you substitute a number into it.

If you are thinking about the shape of a curve then the value of the derivative will tell you if the tangent to the curve at that point has a positive or negative gradient or if the function is increasing or decreasing at that point.

It’s important to make these connections between the derivative which can also be thought of as the gradient or rate of change at a point. We are calling them different things, but they are really all connected.

Here is the function y equals 2x cubed + 3x squared -2. And I’ve drawn the sketch to help us make the connections.

The derivative of the function is dy by dx is 6x squared + 6x.

At the point where x =1, we can substitute this into the derivative and get dy by dx equals 12. So the gradient of the curve at that point is 12.

Now if I was asked to find the equation of the tangent to the curve at that point I could find y and then use y minus b equals m x minus a to find the equation of the tangent.

It also means that the rate of change of the function is 12 at the point where x =1. This is a specific question type. If you are asked to find the rate of change then you differentiate and substitute in the x coordinate.

You could also be asked if the function is increasing or decreasing at the point where x = 1. This question type involves the same process. Substitute x = 1 into the derivative.

As dy by dx equalS 12 the derivative is greater than zero and the function is increasing.

If the derivative had been less that zero then it would have indicated the function was decreasing and equal to zero would have indicated a stationary point.
If you are interested in stationary points then there is another episode to watch that focuses on those.

Earlier we discussed the derivative as a gradient and that we could find the equation of the tangent to the curve. This is a good method to be fluent with.

Here is a different function y equals x squared -4x +7. To find the equation of the tangent to the curve at the point x equals five we first need the gradient.

The derivative at a point gives us the gradient. The derivative is 2x -4. When x = 5, dy by dx is 6 as shown. So the gradient of the tangent is six.

We can find the y coordinate as it isn’t given. We do this by substituting x equals five into the original function. So y is twelve.

Now we use the equation of a straight-line formula to get y = 6x- 18.

So rate of change, gradient and derivative are all connected.
We live in a world where many things change and the application of derivatives are wide.

Make sure you recognise all the different question types that involve the derivative as a gradient or a rate of change.

Understanding change is important in your own life, but maybe even more so in Mathematics. I certainly think so.

This is thinkfour. Thanks for watching.