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.

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

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.