Afgeleide van $$$\cos{\left(b - x \right)}$$$ naar $$$x$$$

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

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:

$${\color{red}\left(\frac{d}{dx} \left(\cos{\left(b - x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{dx} \left(b - x\right)\right)}$$

De afgeleide van de cosinus is $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:

$${\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{dx} \left(b - x\right) = {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{dx} \left(b - x\right)$$

Keer terug naar de oorspronkelijke variabele:

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

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

$$- \sin{\left(b - x \right)} {\color{red}\left(\frac{d}{dx} \left(b - x\right)\right)} = - \sin{\left(b - x \right)} {\color{red}\left(\frac{db}{dx} - \frac{d}{dx} \left(x\right)\right)}$$

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

$$- \left({\color{red}\left(\frac{db}{dx}\right)} - \frac{d}{dx} \left(x\right)\right) \sin{\left(b - x \right)} = - \left({\color{red}\left(0\right)} - \frac{d}{dx} \left(x\right)\right) \sin{\left(b - x \right)}$$

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

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

Dus, $$$\frac{d}{dx} \left(\cos{\left(b - x \right)}\right) = \sin{\left(b - x \right)}$$$.