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cos(x-pi/3)

cos((x - pi)/3)

cos(x - pi/3)

cos(x - pi/3)

Derivative of cos(x-pi/3)

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   /    pi\
cos|x - --|
   \    3 /

$$\cos{\left(x - \frac{\pi}{3} \right)}$$

d /   /    pi\\
--|cos|x - --||
dx\   \    3 //

$$\frac{d}{d x} \cos{\left(x - \frac{\pi}{3} \right)}$$

Transform

Detail solution

Let $u = x - \frac{\pi}{3}$ .
The derivative of cosine is negative sine:

$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - \frac{\pi}{3}\right)$ :
1. Differentiate $x - \frac{\pi}{3}$ term by term:
  1. Apply the power rule: $x$ goes to $1$
  2. The derivative of the constant $- \frac{\pi}{3}$ is zero.
  The result is: $1$
The result of the chain rule is:

$- \sin{\left(x - \frac{\pi}{3} \right)}$
Now simplify:

$\cos{\left(x + \frac{\pi}{6} \right)}$

The answer is:

$\cos{\left(x + \frac{\pi}{6} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    /    pi\
-sin|x - --|
    \    3 /

$$- \sin{\left(x - \frac{\pi}{3} \right)}$$

Simplify

The second derivative [src]

    /    pi\
-sin|x + --|
    \    6 /

$$- \sin{\left(x + \frac{\pi}{6} \right)}$$

Simplify

The third derivative [src]

    /    pi\
-cos|x + --|
    \    6 /

$$- \cos{\left(x + \frac{\pi}{6} \right)}$$

Simplify

The graph