# The definition of the derivative

## Statement

To find the gradient $m$ of any function $f$ at point $x$, you use:

$$m = \lim_{h \to 0}\left(\frac{f(x + h) - f(x)}{h}\right)$$

## The average increase between two points

To get the average increase between points $A$ and $B$, you use:

$$m = \frac{\Delta y}{\Delta x} = \frac{y_A - y_B}{x_A - x_B}$$

When $y = f(x)$, you get the following.

$$m = \frac{f(x_A) - f(x_B)}{x_A - x_B}$$

## The gradient at one point

To get the gradient/slope at one point, we must choose another point that is very close to the first point.

When $x_A$ is the point of which we want to know the gradient/slope, the other point can be $x_A + 0.1$ (the smaller the better). Note that the difference between $-0.01$ and $0.01$ in the denominator is negligible.

$$m = \frac{f(x_A) - f(x_A + 0.1)}{x_A - (x_A + 0.01)} = \frac{f(x_A) - f(x_A + 0.1)}{0.01}$$

The closer the two are to each other, the more accurate the calculation becomes. Yet, we can't use $0$, since dividing by $0$ is undefined.

## Using limits

Although we can't use $0$, we can approach $0$ using limits.

A limit is a way to say that a variable approaches a value while never actually being that value.

$$m = \lim_{h \to 0}\left(\frac{f(x + h) - f(x)}{h}\right)$$

• This is the notation for limits. Write $\lim$ and beneath it the variable and the value it almost but not quite reaches.
• When working with derivatives, we often use $h$ for the almost-zero variable.
• Note that $h$ can't be zero since you'll be dividing by zero then, and that is undefined.

## Notation

The derivative has its own notation. The gradient/slope of $f(x)$ is often written as $f'(x)$. The definition then becomes:

$$f'(x) = \lim_{h \to 0}\left(\frac{f(x + h) - f(x)}{h}\right)$$

This function $f'$ is called "the derivative of $f$", or just "the derivative".

## Proofs building upon this proof

### Chain rule in calculus

This proofs show that the derivative of a nested function is the derivative of the outer function multiplied by the derivative of the inner function.

### Coefficient rule in calculus

This proof shows that the derivative of a function with a coefficient is the coefficient times the derivative of that function.

### Constant rule in calculus

This proofs shows that the derivative of a constant is always zero.

### Product rule in calculus

This proof shows that the derivative of a product g*j is g'*j + g*j'.

### Sum rule in calculus

This proofs shows that the derivative of two functions is the derivative of the first function plus the derivative of the second function.

### Derivative of cos(x)

This proof shows what the derivative of cos(x) is.

### The derivative of e to the x

This proof shows that the definition of e^x is e^x.

### Derivative of sin(x)

This proof shows what the derivative of sin(x) is.