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cos(three /x+p/ four)
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cos3/x+p/4
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cos(3/x-p/4)

The derivative of the function/ cos(3/x+p/4)

Derivative of cos(3/x+p/4)

The solution

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   /3   p\
cos|- + -|
   \x   4/

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

cos(3/x + p/4)

Detail solution

Let $u = \frac{p}{4} + \frac{3}{x}$ .
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 $\frac{\partial}{\partial x} \left(\frac{p}{4} + \frac{3}{x}\right)$ :
1. Differentiate $\frac{p}{4} + \frac{3}{x}$ 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: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
    So, the result is: $- \frac{3}{x^{2}}$
  2. The derivative of the constant $\frac{p}{4}$ is zero.
  The result is: $- \frac{3}{x^{2}}$
The result of the chain rule is:

$\frac{3 \sin{\left(\frac{p}{4} + \frac{3}{x} \right)}}{x^{2}}$
Now simplify:

$\frac{3 \sin{\left(\frac{p}{4} + \frac{3}{x} \right)}}{x^{2}}$

The answer is:

$\frac{3 \sin{\left(\frac{p}{4} + \frac{3}{x} \right)}}{x^{2}}$

The first derivative [src]

     /3   p\
3*sin|- + -|
     \x   4/
------------
      2     
     x

$$\frac{3 \sin{\left(\frac{p}{4} + \frac{3}{x} \right)}}{x^{2}}$$

Simplify

The second derivative [src]

   /                    /3   p\\
   |               3*cos|- + -||
   |     /3   p\        \x   4/|
-3*|2*sin|- + -| + ------------|
   \     \x   4/        x      /
--------------------------------
                3               
               x

$$- \frac{3 \left(2 \sin{\left(\frac{p}{4} + \frac{3}{x} \right)} + \frac{3 \cos{\left(\frac{p}{4} + \frac{3}{x} \right)}}{x}\right)}{x^{3}}$$

Simplify

The third derivative [src]

  /                    /3   p\        /3   p\\
  |               3*sin|- + -|   6*cos|- + -||
  |     /3   p\        \x   4/        \x   4/|
9*|2*sin|- + -| - ------------ + ------------|
  |     \x   4/         2             x      |
  \                    x                     /
----------------------------------------------
                       4                      
                      x

$$\frac{9 \left(2 \sin{\left(\frac{p}{4} + \frac{3}{x} \right)} + \frac{6 \cos{\left(\frac{p}{4} + \frac{3}{x} \right)}}{x} - \frac{3 \sin{\left(\frac{p}{4} + \frac{3}{x} \right)}}{x^{2}}\right)}{x^{4}}$$

Simplify

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Derivative of cos(3/x+p/4)

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