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Derivative of:
Derivative of (x+2)
Derivative of 1+x
Derivative of 1/2*x
Derivative of e^x*cos(x)
Identical expressions
cos((pi*x)/ four -(three *pi)/ four)
co sinus of e of (( Pi multiply by x) divide by 4 minus (3 multiply by Pi ) divide by 4)
co sinus of e of (( Pi multiply by x) divide by four minus (three multiply by Pi ) divide by four)
cos((pix)/4-(3pi)/4)
cospix/4-3pi/4
cos((pi*x) divide by 4-(3*pi) divide by 4)
Similar expressions
cos((pi*x)/4+(3*pi)/4)

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

You have entered [src]

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

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

cos((pi*x)/4 - 3*pi/4)

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

   2    /   /  3   x\\ 
-pi *cos|pi*|- - + -|| 
        \   \  4   4// 
-----------------------
           16

$$- \frac{\pi^{2} \cos{\left(\pi \left(\frac{x}{4} - \frac{3}{4}\right) \right)}}{16}$$

Simplify

The third derivative [src]

  3    /   /  3   x\\
pi *sin|pi*|- - + -||
       \   \  4   4//
---------------------
          64

$$\frac{\pi^{3} \sin{\left(\pi \left(\frac{x}{4} - \frac{3}{4}\right) \right)}}{64}$$

Simplify

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