4.3 The basic rules

The derivative of a constant is 0. If y = f(x) = 6, then $\frac{dy}{dx} = 0$. Ask yourself the question ``how does a constant change as x changes?''

For y = xⁿ, $\frac{dy}{dx} = n\,x^{n-1}.$ "Bring down the power, knock the power down 1."

Examples:

The derivative of x⁴ is $4\,x^3$.

The derivative of x^-2 with respect to x, $\frac{dy}{dx}$, is $-2\, x^{-3}$.

The derivative of a sum is the sum of its derivatives.

Examples

$\frac{d}{dx} (x^2 + x^3) = \frac{d}{dx}x^2 + \frac{d}{dx}x^3 = 2\,x + 3\,x^2$.

$\frac{d}{dx} (4 + x) = \frac{d}{dx} 4 + \frac{d}{dx}x = 1$.

If c is a constant then $\frac{d}{dx} \left(c \times f(x)\right)$ is equal to $c\times \frac{d}{dx} f(x)$.
That, is the derivative passes through a constant multiple.

Examples

$\frac{d}{dx} 5\, x^4 = 5\, \frac{d}{dx}x^4 = 5 \times 4\,x^3 = 20\,x^3$.

The derivative of a straight line must be its slope: $\frac{d}{dx} (m + b\, x) = \frac{d}{dx}m + b\, \frac{d}{dx} x = b$.

For y = e^x the derivative with respect to x, $\frac{dy}{dx}$, is e^x.

For $y = e^{c\,x}$ the derivative with respect to x, $\frac{dy}{dx}$, is $c\, e^{a\,x}$.

For y = ln(x) the derivative with respect to x, $\frac{dy}{dx}$, is $\frac{1}{x}$.