The derivative

On a straight line the slope is the same everywhere. On a curve, the steepness keeps changing. The derivative measures the slope at one point: it is the slope of the tangent there.

How to get it. Take the point P(a, f(a)) and a nearby point Q(a + h, f(a + h)) on the curve.

• The chord PQ has slope [f(a + h) – f(a)]/h.
• Slide Q towards P, that is, let h → 0. The chord turns into the tangent at P.
• So f′(a) = lim (h→0) [f(a + h) – f(a)]/h, if this limit exists.

This is finding the derivative from first principle. Other names for the same thing: the slope at that point, the rate of change of y with respect to x there, and dy/dx.

Example: f(x) = x² at x = 4.

• [(4 + h)² – 16]/h = (8h + h²)/h = 8 + h.
• As h → 0 this tends to 8: the tangent to y = x² at (4, 16) has slope 8.

The derivative function. Do the same at a general point x and you get a formula that works everywhere: f′(x) = lim (h→0) [f(x + h) – f(x)]/h.

• x²: [(x + h)² – x²]/h = 2x + h → 2x. So the slope is 8 at x = 4 and –10 at x = –5.
• x³: [(x + h)³ – x³]/h = 3x² + 3xh + h² → 3x².
• 3x + 2: [3(x + h) + 2 – 3x – 2]/h = 3: the slope of a straight line is the same at every point.
• A constant c: (c – c)/h = 0, so its derivative is 0.
• 1/x: [1/(x + h) – 1/x]/h = –h/[h x(x + h)] = –1/[x(x + h)] → –1/x².

Trig functions from first principle. For cos x, use cos(x + h) = cos x cos h – sin x sin h:

• [cos(x + h) – cos x]/h = cos x × (cos h – 1)/h – sin x × (sin h)/h.
• As h → 0 the first fraction tends to 0 and the second to 1, so (cos x)′ = –sin x.
• The same method gives (sin x)′ = cos x and (tan x)′ = sec²x.