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Identical expressions
7sin^ two ((pi/ sixteen)*t)-3t
7 sinus of squared (( Pi divide by 16) multiply by t) minus 3t
7 sinus of to the power of two (( Pi divide by sixteen) multiply by t) minus 3t
7sin2((pi/16)*t)-3t
7sin2pi/16*t-3t
7sin²((pi/16)*t)-3t
7sin to the power of 2((pi/16)*t)-3t
7sin^2((pi/16)t)-3t
7sin2((pi/16)t)-3t
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7sin^2pi/16t-3t
7sin^2((pi divide by 16)*t)-3t
Similar expressions
7sin^2((pi/16)*t)+3t

The derivative of the function/ 7sin^2((pi/16)*t)-3t

Derivative of 7sin^2((pi/16)*t)-3t

The solution

You have entered [src]

     2/pi  \      
7*sin |--*t| - 3*t
      \16  /

$$- 3 t + 7 \sin^{2}{\left(t \frac{\pi}{16} \right)}$$

7*sin((pi/16)*t)^2 - 3*t

Detail solution

Differentiate $- 3 t + 7 \sin^{2}{\left(t \frac{\pi}{16} \right)}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \sin{\left(t \frac{\pi}{16} \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d t} \sin{\left(t \frac{\pi}{16} \right)}$ :
    1. Let $u = t \frac{\pi}{16}$ .
    2. The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d t} t \frac{\pi}{16}$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $t$ goes to $1$
        
        So, the result is: $\frac{\pi}{16}$
      The result of the chain rule is:
      
      $\frac{\pi \cos{\left(t \frac{\pi}{16} \right)}}{16}$
    The result of the chain rule is:
    
    $\frac{\pi \sin{\left(t \frac{\pi}{16} \right)} \cos{\left(t \frac{\pi}{16} \right)}}{8}$
  So, the result is: $\frac{7 \pi \sin{\left(t \frac{\pi}{16} \right)} \cos{\left(t \frac{\pi}{16} \right)}}{8}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $t$ goes to $1$
  So, the result is: $-3$
The result is: $\frac{7 \pi \sin{\left(t \frac{\pi}{16} \right)} \cos{\left(t \frac{\pi}{16} \right)}}{8} - 3$
Now simplify:

$\frac{7 \pi \sin{\left(\frac{\pi t}{8} \right)}}{16} - 3$

The answer is:

$\frac{7 \pi \sin{\left(\frac{\pi t}{8} \right)}}{16} - 3$

The graph

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Plot the graph f'(x)

The first derivative [src]

             /pi  \    /pi  \
     7*pi*cos|--*t|*sin|--*t|
             \16  /    \16  /
-3 + ------------------------
                8

$$\frac{7 \pi \sin{\left(t \frac{\pi}{16} \right)} \cos{\left(t \frac{\pi}{16} \right)}}{8} - 3$$

Simplify

The second derivative [src]

    2 /   2/pi*t\      2/pi*t\\
7*pi *|cos |----| - sin |----||
      \    \ 16 /       \ 16 //
-------------------------------
              128

$$\frac{7 \pi^{2} \left(- \sin^{2}{\left(\frac{\pi t}{16} \right)} + \cos^{2}{\left(\frac{\pi t}{16} \right)}\right)}{128}$$

Simplify

The third derivative [src]

     3    /pi*t\    /pi*t\
-7*pi *cos|----|*sin|----|
          \ 16 /    \ 16 /
--------------------------
           512

$$- \frac{7 \pi^{3} \sin{\left(\frac{\pi t}{16} \right)} \cos{\left(\frac{\pi t}{16} \right)}}{512}$$

Simplify

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Derivative of 7sin^2((pi/16)*t)-3t

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