The notion of derivative

Differentiation: The derivative

The notion of derivative

The derivative is a function that indicates for each point $x$ what the slope is at that point. In other words, a function that assigns the slope of the tangent line to a point $x$ .

The slope

The slope of a function $\blue{f}$ at a point $a$ can be found by calculating the difference quotient for $[a,a+\orange{h}]$ and letting $\orange{h}$ approach zero. We write this as follows:

$\orange{h}\to0$

If we do not determine the difference quotient at a point but for a variable $x$ , we get the derivative $\blue{f}$ . We denote the derivative with $\green{f'}$ .

For the example on the right, it is only indicated in the second to last step that $\orange{h} \to 0$ . However, this should be at every step, but we have omitted it for the sake of convenience.

Example

$\begin{array}{rcl} \blue{f(x)}&=&\blue{x^2} \\ \green{f'(x)}&=&\dfrac{\blue{(}x+\orange{h}\blue{)^2}-\blue{x^2}}{\orange{h}}\\&=&\dfrac{\blue{x^2}+2x\cdot \orange{h}+\orange{h}^2-\blue{x^2}}{\orange{h}}\\&=&\dfrac{2x\cdot \orange{h}+\orange{h}^2}{\orange{h}}\\&=&2x+\orange{h} \quad \text{with} \quad \orange{h} \to 0\\&=& \green{2x} \end{array}$

We call $f'$ the derivative of $f$ .

The derivative

The derivative of a function $\blue{f}$ is denoted as $f'$ :

$f'(x)=\dfrac{\blue{f(}x+\orange{h}\blue{)}-\blue{f(}x\blue{)}}{\orange{h}} \quad \text{with} \quad \orange{h}\to 0$

Calculating the derivative of a function $f$ is called diferentiation of $f$ .

Not every function can be differentiated. A function of which we can determine the derivative is called a differentiable function. In this course will will only deal with functions that are differentiable.

When we write $\frac{\dd}{\dd x}f$ or $\frac{\dd f}{\dd x}$ , we mean $f'$ ; these three all mean the same thing.

plaatje

We mainly look at the derivation of a function of the form $f(x)=...$ . We can of course also derive functions with another variable, such as $g(t)$ or $h(y)$ . We then get, respectively,

$\begin{array}{rcl}f'(x)&=&\displaystyle\frac{\dd}{\dd x}\left(f(x)\right)\\ g'(t)&=&\displaystyle\frac{\dd}{\dd t}\left(g(t)\right)\\ h'(y)&=&\displaystyle\frac{\dd}{\dd y}\left(h(y)\right)\end{array}$

Hence, we can calculate the derivative by first calculating the difference quotient on an interval with length $\orange{h}$ , and next letting $\orange{h}$ move towards zero. The value which we will obtain by this is the slope of the graph in $x$ . On other words, the slope of the tangent line in the point $x$ .

An example of a non-differentiable function is the absolute value function $\blue{f(x)}=\blue{\abs{x}}$ . This is the function that has value $x$ if $x$ is bigger than or equal to $0$ , and has value $-x$ if $x$ is smaller than $0$ .

This function is non-differentiable in the point $0$ , since we cannot find an unambiguous definition for the tangent line. In the picture we have drawn the graph of the function $\blue{f(x)}=\blue{\abs{x}}$ and two possible tangent lines.

Plaatje

The derivative in a point $a$ is given $f'(a)$ . It is also called the value of the derivative in $a$ , or just the derivative in $a$ . It is the slope of the function in the point $a$

Calculate the derivative of $f(x)=2x^3+9x$ .

$6x^2+9$

For $h \to 0$ , we find:
$\begin{array}{rcl} f'(x)&=&\dfrac{f(x+h)-f(x)}{h}\\ && \blue{\text{definition derivative}}\\ &=& \dfrac{2(x+h)^3 + 9(x+h) - (2x^3+9x)}{h}\\ &&\blue{\text{substituted}}\\ &=& \dfrac{2x^3 + 6x^2\cdot h + 6x\cdot h^2 + 2h^3 + 9x + 9h -2x^3-9x}{h} \\ && \blue{\text{expanded brackets}}\\&=& \dfrac{6x^2\cdot h + 6x\cdot h^2 + 2h^3 +9h }{h} \\&&\blue{2x^3 \text{ and } -2x^3, \text{ and }9x \text{ and } -9x \text{ cancel each other out}}\\ &=& 6x^2 + 6x\cdot h + 2h^2 +9 \\ &&\blue{\text{eliminate }h} \\&=& 6x^2 + 9 \\ && \blue{\text{let }h \text{ approach }0}\end{array}$

New example