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Derivative of cos((pi/3)-4x)

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   /pi      \
cos|-- - 4*x|
   \3       /

$$\cos{\left(- 4 x + \frac{\pi}{3} \right)}$$

d /   /pi      \\
--|cos|-- - 4*x||
dx\   \3       //

$$\frac{d}{d x} \cos{\left(- 4 x + \frac{\pi}{3} \right)}$$

Transform

Detail solution

Let $u = - 4 x + \frac{\pi}{3}$ .
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 $\frac{d}{d x} \left(- 4 x + \frac{\pi}{3}\right)$ :
1. Differentiate $- 4 x + \frac{\pi}{3}$ term by term:
  1. The derivative of the constant $\frac{\pi}{3}$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $4$
    So, the result is: $-4$
  The result is: $-4$
The result of the chain rule is:

$4 \sin{\left(\left(-1\right) 4 x + \frac{\pi}{3} \right)}$
Now simplify:

$4 \cos{\left(4 x + \frac{\pi}{6} \right)}$

The answer is:

$4 \cos{\left(4 x + \frac{\pi}{6} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     /pi      \
4*sin|-- - 4*x|
     \3       /

$$4 \sin{\left(\left(-1\right) 4 x + \frac{\pi}{3} \right)}$$

Simplify

The second derivative [src]

       /      pi\
-16*sin|4*x + --|
       \      6 /

$$- 16 \sin{\left(4 x + \frac{\pi}{6} \right)}$$

Simplify

The third derivative [src]

       /      pi\
-64*cos|4*x + --|
       \      6 /

$$- 64 \cos{\left(4 x + \frac{\pi}{6} \right)}$$

Simplify

The graph