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,

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

In words 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 value of the derivative 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)=3x^3+3x#.

#9x^2+3#

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

New example