Afgeleide van $$$\left(x - 1\right)^{2}$$$

De functie $$$\left(x - 1\right)^{2}$$$ is de samenstelling $$$f{\left(g{\left(x \right)} \right)}$$$ van twee functies $$$f{\left(u \right)} = u^{2}$$$ en $$$g{\left(x \right)} = x - 1$$$.

Pas de kettingregel $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ toe:

$${\color{red}\left(\frac{d}{dx} \left(\left(x - 1\right)^{2}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{2}\right) \frac{d}{dx} \left(x - 1\right)\right)}$$

Pas de machtsregel $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ toe met $$$n = 2$$$:

$${\color{red}\left(\frac{d}{du} \left(u^{2}\right)\right)} \frac{d}{dx} \left(x - 1\right) = {\color{red}\left(2 u\right)} \frac{d}{dx} \left(x - 1\right)$$

Keer terug naar de oorspronkelijke variabele:

$$2 {\color{red}\left(u\right)} \frac{d}{dx} \left(x - 1\right) = 2 {\color{red}\left(x - 1\right)} \frac{d}{dx} \left(x - 1\right)$$

De afgeleide van een som/verschil is de som/het verschil van de afgeleiden:

$$2 \left(x - 1\right) {\color{red}\left(\frac{d}{dx} \left(x - 1\right)\right)} = 2 \left(x - 1\right) {\color{red}\left(\frac{d}{dx} \left(x\right) - \frac{d}{dx} \left(1\right)\right)}$$

Pas de machtsregel $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ toe met $$$n = 1$$$, met andere woorden, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$2 \left(x - 1\right) \left({\color{red}\left(\frac{d}{dx} \left(x\right)\right)} - \frac{d}{dx} \left(1\right)\right) = 2 \left(x - 1\right) \left({\color{red}\left(1\right)} - \frac{d}{dx} \left(1\right)\right)$$

De afgeleide van een constante is $$$0$$$:

$$2 \left(1 - {\color{red}\left(\frac{d}{dx} \left(1\right)\right)}\right) \left(x - 1\right) = 2 \left(1 - {\color{red}\left(0\right)}\right) \left(x - 1\right)$$

Dus, $$$\frac{d}{dx} \left(\left(x - 1\right)^{2}\right) = 2 x - 2$$$.