Lesson 3 / 5 · 14 min read

Derivatives

The slope of a curve at a point — and the rules to compute it.

What is a derivative?

The derivative of $f$ at $x$ is the slope of the tangent line to $f$ at that point. Formally:

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

You almost never compute it from this definition — there are shortcuts.

For any real number $n$ :

$\frac{d}{dx}\left[x^n\right] = n x^{n-1}$

Examples.

$\frac{d}{dx}[x^5] = 5x^4$
$\frac{d}{dx}[x] = 1$ (since $n = 1$ )
$\frac{d}{dx}[\sqrt{x}] = \frac{d}{dx}[x^{1/2}] = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$
$\frac{d}{dx}[7] = 0$ (the derivative of any constant is 0)

You can pull constants out, and the derivative of a sum is the sum of derivatives:

$\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$ $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$

Example. $f(x) = 3x^4 - 2x^2 + 5x - 9$ .

$f'(x) = 12x^3 - 4x + 5$

If you have a product $f \cdot g$ :

$(fg)' = f'g + fg'$

Example. $h(x) = x^2 \sin(x)$ . Then $h'(x) = 2x \sin(x) + x^2 \cos(x)$ .

$\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$

For composite functions $f(g(x))$ :

$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

Example. $\frac{d}{dx} \sin(x^2) = \cos(x^2) \cdot 2x$ .