Derivatives of polynomials and power functions

Optimization: Differentiation

Derivatives of polynomials and power functions

It is often necessary to calculate the minimum or maximum (in short: an extremum) of a function. For example, to optimize the manufacturing costs of a product depending on the percentage of a specific raw material.

If the graph of a function has a tangent line at every point, then that tangent will be horizontal at an extremum. This means that the slope of the tangent to the graph of the function at such a point is equal to #0#. To calculate the minimum and/or maximum, we will therefore use an auxiliary function: the function that gives the slope of the tangent line through that point at each point of the function. This function is called the derivative.

We will first look at what the derivative is and then indicate how we can calculate that derivative in a number of common cases.

Differentiation

Let #f(x)# be a function on an interval #I#. The derivative function #f'(x)# is the function that at each point #\rv{x, f(x)}# gives the slope at that point. So #f'(x)# is the slope of the graph of #f# at the point #\rv{x,f(x)}#; that means it is the slope of the tangent line to the graph of #f(x)# at that point.

If #f# is differentiable at all points of an interval #I#, that is, we can calculate the derivative function at all points, then we say that #f# is differentiable on #I#. In that case #f’# is a function on #I#.

The value #f'(x)# is often referred to as #\frac{{\dd}}{{\dd}x}f(x)# and is called the derivative of #f# in #x#.

If #y# is a function rule of #f# (the value of #f# at any point #x#) and #a# is a value for #x# where #f(x)# is defined, then we also write #\left.\frac{{\dd }}{{\dd}x}y\right|_{x=a}# instead of #f'(a)# or #\frac{{\dd y }}{{\dd}x}(a)#.

The function #f# and the function statement #f(x)# are often used interchangeably. The expressions #f'(x)# and #\dfrac{\dd}{\dd x}f(x)# are also used instead of #f'#.

An example of using the vertical bar with #f(x)=x^2+1#, where #f'(x) = 2x# (as we will see below), is the following calculation of the derivative of #f# in #3# : \[\frac{\dd f}{\dd x}(3)=\left.\frac{\dd}{\dd x}(x^2+1)\right|_{x=3}=\left.(2x)\right|_{x=3}=6\]

Example

Take a look at the the graph of function #f(x)=x^2-x+5# (in blue) and the tangent line to it (in red) drawn at the point #\rv{2,7}#.

The tangent line intersects the graph only at that point. Also, the tangent line lies either below or above the graph around that specific point (in other words, it lies on one side of the graph).

The derivative of #f(x)# at #x=2# is equal to #3#. This is then the slope of the tangent line.

New example

In general, not every function is differentiable. For example, in #0#, the function #f(x) = |x|# (the absolute value of #x#) is not differentiable. The graph has no clear tangent for #x=0#. This will not be discussed in more detail in this course.

Before we look at how to differentiate polynomials and power functions, let's first look at three useful basic rules for differentiation.

Three basic rules for differentiation

Let #c# be a real number.

Constant rule: The derivative of the constant function #f(x)=c# is #f'(x)=0#.
Product-with-constant rule: The derivative of the product #c\cdot f(x)# of the constant #c# with a function #f# is #c\cdot f'(x)#.
Sum rule: If #f# and #g# are functions, then the derivative of the sum function #f(x)+g(x)# is equal to #f'(x)+g'(x)#.

The derivative of the constant function is zero. After all, the graph of this function is a horizontal line.
The equation of the tangent line to the graph of #f# at the point #\rv{a,b}# of the graph is #y = b+f'(a) \cdot(x-a)#. The point on the graph of #c\cdot f(x)# with the same #x#-coordinate #a# is then #\rv{a,c\cdot b}# and the equation for the tangent through that point to the graph of #c\cdot f'# is then #y=c\cdot b+c\cdot f'(a)\cdot(x-a)#. The slope of the tangent is therefore #\left.\frac{\dd}{c\cdot f'(x)}\right|_a = c\cdot f'(a)#. The conclusion is that the derivative of #c\cdot f(x)# is equal to #c\cdot f'(x)#.
If #y = b+f'(a) \cdot(x-a)# is the tangent to the graph of #f# at the point #\rv{a,b}# and if #y = c+g'(a) \cdot(x-a)# is the tangent to the graph of #g# at the point #\rv{a,c}#, then #y =b+c+ \left(f'(a)+g'(a)\right)\cdot(x-a)# is the tangent to the graph of #f(x)+g(x) # at the point #\rv{a,b+c}#. This shows that #\frac{\dd}{\dd x}\left(f(x)+g(x)\right) = \frac{\dd}{\dd x}f(x) +\frac{\dd}{\dd x}g(x)#.

The linear function #3x+4# has #3# as its derivative. After all, that is the slope of the graph of #f#, which is itself a line. But if we use the fact that the derivative of the function #x# is equal to #1#, then this fact also follows from applying the above rules:

\[\begin{array}{rcl} f'(x) &=&\frac{\dd}{\dd x}\left(3x+4\right)\\ &=&\frac{\dd}{\dd x}\left(3x\right)+\frac{\dd}{\dd x}\left(4\right)\\ &&\phantom{xx}\color{blue}{\text{sum rule}}\\ &=&\frac{\dd}{\dd x}\left(3x\right)+0\\ &&\phantom{xx}\color{blue}{\text{constant rule}}\\ &=&3\frac{\dd}{\dd x}\left(x\right)\\ &&\phantom{xx}\color{blue}{\text{product-with-constant-rule}}\\ &=&3\\ &&\phantom{xx}\color{blue}{\frac{\dd}{\dd x}\left(x\right)=1}\end{array}\]

In order to differentiate polynomials, we still need to get to know the derivative of the power functions #x^n#, where #n# is a natural number. This is a special case of the first rule below, which deals with the derivative of all real power functions

Power rule for differentiation

If #a# is a real number, then the derivative of the function #x^a# is #a\cdot x^{a-1}#. In other words: \[\frac{\dd}{{\dd}x}x^a = a\cdot x^{a-1}\]

A consequence of this is the polynomial rule: If #n# is a natural number and #a_0#, #a_1,\ldots, a_n# are real numbers, then
\[\frac{\dd}{{\dd}x}\left(a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0\right)=n\cdot a_nx^{n-1}+(n-1)\cdot a_{n-1}x^{n-2}+\cdots+a_1\]

For #a=0#, #x^a# is a constant function, whose derivative is already known to be equal to #0#.

With the above rule and with the sum rule, the derivative of every polynomial function \(a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0\) can be found. This can be seen through repeated application of the three basic rules of differentiation.

For #a=0#, #x^a# is a constant function, whose derivative is already known to be equal to #0#. For #a=1# the graph of #x^a# is a line, of which we already know that the derivative is the constant function #1#. We omit the evidence for other values of #a#.

The second statement, about the derivative of a polynomial function, can be proven by using the first statement and repeatedly applying the three basic rules of differentiation.

Calculate the derivative of #y=# #5 x ^4#.

#y'=# #20 x ^3#

Using the power rule for differentiation #\frac{\dd}{\dd x}\left(a\cdot x^n\right) = n\cdot a\cdot x^{n-1}# with #a=5# and #n=4#, we find: \[ y'=\frac{\dd}{\dd x}y=\frac{\dd}{\dd x}\left(5 x ^4\right) =5 \cdot 4 \cdot x^{3} =20 x ^3 \]

New example