Chapter 8 - The Derivative of a Polynomial

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Lesson Summary: 
In this lesson we tell you the derivative of a polynomial and give you a tool for calculating its derivative both symbolically and numerically at any point.
Lesson Inputs: 
Chapter 7 - The Tangent Line
Lesson Outputs: 
Chapter 9 - The Product Rule
Lesson: 

One of the easiest formulas to memorize is the derivative of a polynomial. Recall that the form of a polynomial is
\[
f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0.
\]
Its derivative is:

The Derivative of a Polynomial
If $f(x)$ is the polynomial $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, then its derivative is
\[
f'(x)=n\,a_nx^{n-1}+(n-1)\,a_{n-1}x^{n-2}+\cdots+a_1.
\]

You really don't have to memorize this rule because it comes from our first four rules of derivatives.

Let's do a quick example by finding the derivative of

\[f(x)=6x^4+7x^2+8x+5.\]

Using the rule above we know that

\begin{eqnarray}f\,'(x)&=&4\cdot 6\cdot x^{4-1}\nonumber\\&+&2\cdot 7\cdot x^{2-1}\nonumber\\&+&1\cdot 8\cdot x^{1-1}\nonumber\\&+&0\cdot 5\cdot x^{0-1}\nonumber\\&=&24x^3+14x+8.\nonumber\end{eqnarray}

In the calculation we viewed $5$ as $5x^0$ because $x^0=1$. It's derivative is $0\cdot 5\cdot x^{-1}=(0\cdot 5)/x=0$. In other words, the Power Rule is consistent with the rule that the derivative of a constant is zero.

@interact
def _(f=input_box(x^2,width=40,label='f(x)='), c=input_box(1,width=5,label='Derivative at x='),z=input_box(5,width=5,label='Zoom:')):
z=abs(z)
domain=[x,c-z,c+z]
fc=f(x=c)
df=f.diff(x)
dfc=df(x=c)
tl=fc+(x-c)*dfc
show('f(x)='+latex(f)+'\mbox{ and }f\'(x)='+latex(df))
cn=N(c,digits=4)
fcn=N(fc,digits=4)
dfcn=N(dfc,digits=4)
show('f('+latex(cn)+')='+latex(fcn)+'\mbox{ and }f\'('+latex(cn)+')='+latex(dfcn))
p=plot(f,domain,thickness=2,color="#0099cc")+plot(tl,domain,thickness=2,color="#cc0000")
p.axes_width(1)
p.axes_color((0.3,0.3,0.3))
p.tick_label_color((0.3,0.3,0.3))
p.set_legend_options(loc=(0.15,0.65))
p.legend(false)
p.show(gridlines=True,aspect_ratio='automatic',figsize=5,ymin=fc-z,ymax=fc+z)

Exercise: Use the tool above to determine the derivatives of the following functions:

  • $f(x)=5x^7+16x^3-9$ at $x=4$
  • $f(x)=x^2$ at $x=-2.3$
  • $f(x)=-2.3x^3-5.2$ at $x=-2$
  • $f(x)=16x^3-8x^2+15x-22$ at $x=10$

The blue line is the function and the red line is the tangent line. Use "*" for multiplication and "^" for exponentiation. For example, to do the first problem you would input 5*x^7+16*x^3-9 into the "f(x)=" box and then 4 in the "x=" box. Try it now!