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Derivative of sin(x/5-pi/3)

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   /x   pi\
sin|- - --|
   \5   3 /

$$\sin{\left(\frac{x}{5} - \frac{\pi}{3} \right)}$$

sin(x/5 - pi/3)

Detail solution

Let $u = \frac{x}{5} - \frac{\pi}{3}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\frac{x}{5} - \frac{\pi}{3}\right)$ :
1. Differentiate $\frac{x}{5} - \frac{\pi}{3}$ term by term:
  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: $\frac{1}{5}$
  2. The derivative of the constant $- \frac{\pi}{3}$ is zero.
  The result is: $\frac{1}{5}$
The result of the chain rule is:

$\frac{\cos{\left(\frac{x}{5} - \frac{\pi}{3} \right)}}{5}$
Now simplify:

$\frac{\sin{\left(\frac{x}{5} + \frac{\pi}{6} \right)}}{5}$

The answer is:

$\frac{\sin{\left(\frac{x}{5} + \frac{\pi}{6} \right)}}{5}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   /x   pi\
cos|- - --|
   \5   3 /
-----------
     5

$$\frac{\cos{\left(\frac{x}{5} - \frac{\pi}{3} \right)}}{5}$$

Simplify

The second derivative [src]

    /-5*pi + 3*x\ 
-sin|-----------| 
    \     15    / 
------------------
        25

$$- \frac{\sin{\left(\frac{3 x - 5 \pi}{15} \right)}}{25}$$

Simplify

The third derivative [src]

    /-5*pi + 3*x\ 
-cos|-----------| 
    \     15    / 
------------------
       125

$$- \frac{\cos{\left(\frac{3 x - 5 \pi}{15} \right)}}{125}$$

Simplify