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Derivative of:
Derivative of sin(4x)
Derivative of tan(4*x)+2/3*tan(4*x)^(3)+1/5*tan(4*x)^(5)
Derivative of log(x+5)-2*x+9
Derivative of ((ln(x))^2)/x
Identical expressions
y= two sin(5x/2+2п/ three)
y equally 2 sinus of (5x divide by 2 plus 2п divide by 3)
y equally two sinus of (5x divide by 2 plus 2п divide by three)
y=2sin5x/2+2п/3
y=2sin(5x divide by 2+2п divide by 3)
Similar expressions
y=2sin(5x/2-2п/3)

The derivative of the function/ y=2sin(5x/2+2п/3)

Derivative of y=2sin(5x/2+2п/3)

The solution

You have entered [src]

     /5*x   2*pi\
2*sin|--- + ----|
     \ 2     3  /

$$2 \sin{\left(\frac{5 x}{2} + \frac{2 \pi}{3} \right)}$$

2*sin((5*x)/2 + (2*pi)/3)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \frac{5 x}{2} + \frac{2 \pi}{3}$ .
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 $\frac{d}{d x} \left(\frac{5 x}{2} + \frac{2 \pi}{3}\right)$ :
  1. Differentiate $\frac{5 x}{2} + \frac{2 \pi}{3}$ 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.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $5$
      So, the result is: $\frac{5}{2}$
    2. The derivative of the constant $\frac{2 \pi}{3}$ is zero.
    The result is: $\frac{5}{2}$
  The result of the chain rule is:
  
  $\frac{5 \cos{\left(\frac{5 x}{2} + \frac{2 \pi}{3} \right)}}{2}$
So, the result is: $5 \cos{\left(\frac{5 x}{2} + \frac{2 \pi}{3} \right)}$
Now simplify:

$- 5 \sin{\left(\frac{5 x}{2} + \frac{\pi}{6} \right)}$

The answer is:

$- 5 \sin{\left(\frac{5 x}{2} + \frac{\pi}{6} \right)}$

The graph

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Plot the graph f'(x)

The first derivative [src]

     /5*x   2*pi\
5*cos|--- + ----|
     \ 2     3  /

$$5 \cos{\left(\frac{5 x}{2} + \frac{2 \pi}{3} \right)}$$

Simplify

The second derivative [src]

       /4*pi + 15*x\
-25*sin|-----------|
       \     6     /
--------------------
         2

$$- \frac{25 \sin{\left(\frac{15 x + 4 \pi}{6} \right)}}{2}$$

Simplify

The third derivative [src]

        /4*pi + 15*x\
-125*cos|-----------|
        \     6     /
---------------------
          4

$$- \frac{125 \cos{\left(\frac{15 x + 4 \pi}{6} \right)}}{4}$$

Simplify