Basic Derivatives

Definition of the derivative

The derivative of the function $f$ with respect to the variable $x$ is the function $f^{'}$ whose value at $x$ is $f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Slope of a curve at a point

The slope of the curve $y = f (x)$ at the point $(a, f (a))$ is $m = lim_{h \to 0} \frac{f (a + h) - f (a)}{h}$
- (This is also the equation for the slope of the tangent to $y = f (x)$ at that point)
The normal line to a curve at a point is the line perpendicular to a tangent at that point.

Shortcut for polynomial derivatives

If you have a polynomial of the form $y = x^{n}$ , its derivative is $y^{'} = n x^{n - 1}$ for each term of the polynomial. This is good for verification but limit notation should still be used in work.

Example

For $x^{3} - 2 x^{2} + 3 x - 4$ , the derivative comes out to be (term-by-term) $y^{'} = 3 x^{2} - 4 x^{1} + 3 x^{0} - 0$ , simplifying to $y^{'} = 3 x^{2} + 4 x + 3$ .