Differentiating Powers
Differentiating Quotients
UBC Calculus Online Course Notes

Differentiating Polynomials

Remember that a polynomial is formed by summing terms of the form $  cx^n 
 $ where $  c  $ is a constant multiplier and $  n  $ is a non-negative integer.



Example:

An example is $  2x^3 - x  + 3.  $ This may be easily differentiated using the rules we have so far:


\begin{eqnarray*} 
  \frac{d}{dx}(2x^3 - x + 3) & = &\frac{d}{dx}(2x^3) + \frac{d}{dx}(-1 
      x) + \frac{d}{dx}(3) \\ 
  & = & 2\frac{d}{dx}x^3 - \frac{d}{dx} x + \frac{d}{dx} 3 \\ 
  & = & 6x^2 - 1 + 0 \\ 
  & = & 6x^2 - 1. 
\end{eqnarray*}


where we have first applied the sum rule, then the constant multiplier rule and finally the power rule.

Below, the graph of the function $  y(x) = 2x^3 - x + 3 
 $ is shown on the left while its derivative $  6x^2 - 1 
 $ is shown on the right.



Questions:

  1. Can you explain the relationship between these two graphs?

  2. What does it mean when the derivative is negative and what does it mean when it is positive?

  3. What happens at the points where the derivative is equal to zero?