Differentiating polynomials

What differentiation does

• The gradient measures steepness. On a straight line it is constant, but on a curve it changes from point to point.
• Differentiation finds the gradient function (also called the derivative) — a new expression that gives the gradient of the curve at any value of $x$.
• Notation: the derivative of $y$ is written $\dfrac{dy}{dx}$ (read "dee y by dee x"); for a function $f(x)$ it is written $f'(x)$ ("f dash of x"). They mean the same thing.

The power rule

• To differentiate a power of $x$: bring the power down to multiply, then reduce the power by 1:

$\text{if } y = x^n \text{ then } \frac{dy}{dx} = n\,x^{\,n-1}$

• A coefficient (number in front) is kept and multiplied through: $\dfrac{d}{dx}(a x^n) = an\,x^{n-1}$.
• So $\dfrac{d}{dx}(3x^4) = 3 \times 4\,x^3 = 12x^3$.

Differentiating a whole polynomial

• Differentiate each term separately and keep the $+$/$-$ signs between them.
• The derivative of a plain number (constant) is 0 — a constant is a flat, horizontal line with zero gradient.
• The derivative of $ax$ (i.e. $ax^1$) is just $a$, because the power 1 drops to $x^0 = 1$.

Seeing the gradient function on a graph

• The derivative is itself a function, so it has its own graph. Putting the two graphs side by side shows exactly what the "gradient function" is doing.
• Take the curve $y = x^3 - 3x$. Differentiating gives the gradient function $\dfrac{dy}{dx} = 3x^2 - 3$.

• Read the two graphs together, lining up the same $x$-values (the dashed lines join the matching points):
  • Where the curve is flat — its turning points at $x = -1$ and $x = 1$ — the gradient is zero, so the gradient graph crosses the $x$-axis at exactly those $x$-values.
  • Where the curve slopes down (between $x = -1$ and $x = 1$), the gradient is negative, so the gradient graph sits below the $x$-axis.
  • Where the curve slopes up ($x < -1$ and $x > 1$), the gradient is positive, so the gradient graph sits above the $x$-axis.
  • The steeper the curve, the further from zero the value on the gradient graph.
• So the gradient function is a map of the original curve's steepness: read off any $x$, and it tells you the slope of the curve at that point.