Most people know that speed is the rate of change of distance over time, i.e. \[ \text{speed} = \frac{\text{distance}}{\text{time}} \ \]
To be more precise, if the distance at time t0 is x0, and the distance at time t1 is x1, then the average speed is calculated as \[ \text{average speed} = \frac{x_1-x_0}{t_1-t_0} \quad \ \]
As the difference between time intervals decreases, the expression becomes \[ \text{speed} = \frac{\Delta x}{\Delta t} \] where the \( \Delta \) symbol means 'a very small difference'.
The diagram above is a plot of distance as a function of time. As \( \Delta x \) and \( \Delta t \) become smaller, the ratio \[ \frac{\Delta x}{\Delta t} \quad \ \] more and more closely approximates the gradient of the curve at the point t0. So the gradient of the curve gives the value of the instantaneous speed.
So how do you calculate the gradient of a curve?
A secant is a line passing through two points on a curve.
The gradient of the secant PQ in the diagram is just the differences in the y-coordinates divided by the differences in the x-coordinates. \[ \text{gradient of PQ} = \frac{\Delta y}{\Delta x} = \frac{y_2-y_1}{x_2-x_1} \quad \ \]
A tangent is a line which shares a common point with the curve but does not intersect the curve at that common point (but may intersect the curve at some other point). The tangent is the limiting position of a secant as the differences in the x-coordinates of the two points of the secant become smaller and smaller.
As the point c gets closer to x, the secant gets closer to the tangent.
The gradient of the tangent at a point is called the derivative of the function at the point, or sometimes the differential coefficient.
The derivative formalises the idea of being the limiting value of \[ \frac{\Delta y}{\Delta x} \quad \ \]
Derivatives are all about approximating curves at points of interest by straight lines, mainly because straight lines are easier to calculate with, and also because, provided you stay 'close enough' to the point of interest, the approximation is a good one.
Guided Examples
O E Qs
The diagram below shows the general situation.
The definition of the derivative is \[ \begin{align} f\ '(x) &= \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} \\ &= \lim_{h \to 0} \frac{f(x+h) - f(x)}{x + h - x} \\ &= \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \end{align} \ \]
This definition matches the idea of a secant becoming a tangent.
The expression f(x+h) means to substitute (x+h) whenever you see x in the definition of the function. So given \[ f(x) = x^2+x+1\] then \[ f(x+h) = (x+h)^2+(x+h)+1\]
Consider f(x) = c. Here c is a constant. There is no 'x' in the right hand side of the definition of this function. In this case, f(x+h) = c. Substitute this expression into the definition to calculate the derivative of a constant.
Repeat the above for f(x) = -1.
Consider f(x) = x. Here we have f(x+h) = x+h. Substitute this expression into the definition to calculate the derivative.
The real power of the definition is that you can use it to calculate the derivative of a curve at any point.
Calculate the derivative of y = x2 + 3.
Example To calculate the derivative of y = x2 + 3x, \[ \begin{align} f\ '(x) &= \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \\ &= \lim_{h \to 0} \frac{(x+h)^2+3(x+h) - (x^2+3x)}{h} \\ &= \lim_{h \to 0} \frac{x^2+2xh+h^2+3x+3h-x^2-3x}{h} \\ &= \lim_{h \to 0} \frac{2xh+h^2+3h}{h} \\ &= \lim_{h \to 0} \frac{h(2x+h+3)}{h} \\ &= 2x+3 \end{align} \ \] This is a bit clumsy. Later lessons will provide 'formulas' for derivatives.
The derivative of a function at a point x is denoted by \[ f\ '(x) \quad \text{ or } \quad \frac{dy}{dx} \ \]
As you can see from the calculations above, the result of calculating the derivative is another function.
Calculate the derivative at a point by substituting the value of the point into the expression for the derivative.
Example If \[ \begin{align} f\ '(x) &= 2x+3 \end{align} \ \] then the value of the derivative at x = 3 is \[ \begin{align} f\ '(3) &= 2(3)+3 = 9 \\ \end{align} \ \]
Consider f(x) = x. Calculate the value of the derivative at the point x = 2.
Another notation is \[ \begin{align} \frac{dy}{dx} \bigg |_{x=a} \end{align} \ \]
If a function is differentiable at every point in its domain it is called differentiable.
If a function is differentiable then it is continuous. Note - the converse of this may be false, but many functions which are continuous are also differentiable - these are the functions which have no 'kinks' in them.