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Derivative of:
Derivative of y=5
Derivative of x^3*lnx
Derivative of x^3+1/x-1
Derivative of x^2/cosx
Identical expressions
y=[sin5x]-(one / three)[sin^ three ][5x]
y equally [ sinus of 5x] minus (1 divide by 3)[ sinus of cubed ][5x]
y equally [ sinus of 5x] minus (one divide by three)[ sinus of to the power of three ][5x]
y=[sin5x]-(1/3)[sin3][5x]
y=[sin5x]-1/3[sin3][5x]
y=[sin5x]-(1/3)[sin³][5x]
y=[sin5x]-(1/3)[sin to the power of 3][5x]
y=[sin5x]-1/3[sin^3][5x]
y=[sin5x]-(1 divide by 3)[sin^3][5x]
Similar expressions
y=[sin5x]+(1/3)[sin^3][5x]

The derivative of the function/ y=[sin5x]-(1/3)[sin^3][5x]

Derivative of y=[sin5x]-(1/3)[sin^3][5x]

The solution

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              3       
           sin (x)    
sin(5*x) - -------*5*x
              3

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

sin(5*x) - sin(x)^3/3*5*x

Detail solution

Differentiate $- 5 x \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(5 x \right)}$ term by term:
1. Let $u = 5 x$ .
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 $\frac{d}{d x} 5 x$ :
  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: $5$
  The result of the chain rule is:
  
  $5 \cos{\left(5 x \right)}$
4. 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 product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Apply the power rule: $x$ goes to $1$
      $g{\left(x \right)} = \sin^{3}{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
      1. Let $u = \sin{\left(x \right)}$ .
      2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
        
        The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
        
        The result of the chain rule is:
        
        $3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
      The result is: $3 x \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \sin^{3}{\left(x \right)}$
    So, the result is: $5 x \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \frac{5 \sin^{3}{\left(x \right)}}{3}$
  So, the result is: $- 5 x \sin^{2}{\left(x \right)} \cos{\left(x \right)} - \frac{5 \sin^{3}{\left(x \right)}}{3}$
The result is: $- 5 x \sin^{2}{\left(x \right)} \cos{\left(x \right)} - \frac{5 \sin^{3}{\left(x \right)}}{3} + 5 \cos{\left(5 x \right)}$

The answer is:

$- 5 x \sin^{2}{\left(x \right)} \cos{\left(x \right)} - \frac{5 \sin^{3}{\left(x \right)}}{3} + 5 \cos{\left(5 x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                  3                        
             5*sin (x)          2          
5*cos(5*x) - --------- - 5*x*sin (x)*cos(x)
                 3

$$- 5 x \sin^{2}{\left(x \right)} \cos{\left(x \right)} - \frac{5 \sin^{3}{\left(x \right)}}{3} + 5 \cos{\left(5 x \right)}$$

Simplify

The second derivative [src]

  /                   3           2                    2          \
5*\-5*sin(5*x) + x*sin (x) - 2*sin (x)*cos(x) - 2*x*cos (x)*sin(x)/

$$5 \left(x \sin^{3}{\left(x \right)} - 2 x \sin{\left(x \right)} \cos^{2}{\left(x \right)} - 2 \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 5 \sin{\left(5 x \right)}\right)$$

Simplify

The third derivative [src]

  /                    3           2                    3             2          \
5*\-25*cos(5*x) + 3*sin (x) - 6*cos (x)*sin(x) - 2*x*cos (x) + 7*x*sin (x)*cos(x)/

$$5 \left(7 x \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 2 x \cos^{3}{\left(x \right)} + 3 \sin^{3}{\left(x \right)} - 6 \sin{\left(x \right)} \cos^{2}{\left(x \right)} - 25 \cos{\left(5 x \right)}\right)$$

Simplify

The graph

Derivative of y=[sin5x]-(1/3)[sin^3][5x]

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Identical expressions

Similar expressions

Derivative of y=[sin5x]-(1/3)[sin^3][5x]

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Piecewise:

The solution