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Identical expressions
fifty *sin(y*cos(y))^(three)
50 multiply by sinus of (y multiply by co sinus of e of (y)) to the power of (3)
fifty multiply by sinus of (y multiply by co sinus of e of (y)) to the power of (three)
50*sin(y*cos(y))(3)
50*siny*cosy3
50sin(ycos(y))^(3)
50sin(ycos(y))(3)
50sinycosy3
50sinycosy^3

The derivative of the function/ 50*sin(y*cos(y))^(3)

Derivative of 50sin(ycos(y))^(3)

The solution

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      3          
50*sin (y*cos(y))

$$50 \sin^{3}{\left(y \cos{\left(y \right)} \right)}$$

50*sin(y*cos(y))^3

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \sin{\left(y \cos{\left(y \right)} \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d y} \sin{\left(y \cos{\left(y \right)} \right)}$ :
  1. Let $u = y \cos{\left(y \right)}$ .
  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 $\frac{d}{d y} y \cos{\left(y \right)}$ :
    1. Apply the product rule:
      
      $\frac{d}{d y} f{\left(y \right)} g{\left(y \right)} = f{\left(y \right)} \frac{d}{d y} g{\left(y \right)} + g{\left(y \right)} \frac{d}{d y} f{\left(y \right)}$
      
      $f{\left(y \right)} = y$ ; to find $\frac{d}{d y} f{\left(y \right)}$ :
      1. Apply the power rule: $y$ goes to $1$
      $g{\left(y \right)} = \cos{\left(y \right)}$ ; to find $\frac{d}{d y} g{\left(y \right)}$ :
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d y} \cos{\left(y \right)} = - \sin{\left(y \right)}$
      The result is: $- y \sin{\left(y \right)} + \cos{\left(y \right)}$
    The result of the chain rule is:
    
    $\left(- y \sin{\left(y \right)} + \cos{\left(y \right)}\right) \cos{\left(y \cos{\left(y \right)} \right)}$
  The result of the chain rule is:
  
  $3 \left(- y \sin{\left(y \right)} + \cos{\left(y \right)}\right) \sin^{2}{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)}$
So, the result is: $150 \left(- y \sin{\left(y \right)} + \cos{\left(y \right)}\right) \sin^{2}{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)}$

The answer is:

$150 \left(- y \sin{\left(y \right)} + \cos{\left(y \right)}\right) \sin^{2}{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2                                             
150*sin (y*cos(y))*(-y*sin(y) + cos(y))*cos(y*cos(y))

$$150 \left(- y \sin{\left(y \right)} + \cos{\left(y \right)}\right) \sin^{2}{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)}$$

Simplify

The second derivative [src]

     /                    2    2                                   2    2                                                              \              
-150*\(-cos(y) + y*sin(y)) *sin (y*cos(y)) - 2*(-cos(y) + y*sin(y)) *cos (y*cos(y)) + (2*sin(y) + y*cos(y))*cos(y*cos(y))*sin(y*cos(y))/*sin(y*cos(y))

$$- 150 \left(\left(y \sin{\left(y \right)} - \cos{\left(y \right)}\right)^{2} \sin^{2}{\left(y \cos{\left(y \right)} \right)} - 2 \left(y \sin{\left(y \right)} - \cos{\left(y \right)}\right)^{2} \cos^{2}{\left(y \cos{\left(y \right)} \right)} + \left(y \cos{\left(y \right)} + 2 \sin{\left(y \right)}\right) \sin{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)}\right) \sin{\left(y \cos{\left(y \right)} \right)}$$

Simplify

The third derivative [src]

    /                        3    3                2                                                       3                                                                              3    2                                2                                                                   \
150*\- 2*(-cos(y) + y*sin(y)) *cos (y*cos(y)) + sin (y*cos(y))*(-3*cos(y) + y*sin(y))*cos(y*cos(y)) - 3*sin (y*cos(y))*(-cos(y) + y*sin(y))*(2*sin(y) + y*cos(y)) + 7*(-cos(y) + y*sin(y)) *sin (y*cos(y))*cos(y*cos(y)) + 6*cos (y*cos(y))*(-cos(y) + y*sin(y))*(2*sin(y) + y*cos(y))*sin(y*cos(y))/

$$150 \left(\left(y \sin{\left(y \right)} - 3 \cos{\left(y \right)}\right) \sin^{2}{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)} + 7 \left(y \sin{\left(y \right)} - \cos{\left(y \right)}\right)^{3} \sin^{2}{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)} - 2 \left(y \sin{\left(y \right)} - \cos{\left(y \right)}\right)^{3} \cos^{3}{\left(y \cos{\left(y \right)} \right)} - 3 \left(y \sin{\left(y \right)} - \cos{\left(y \right)}\right) \left(y \cos{\left(y \right)} + 2 \sin{\left(y \right)}\right) \sin^{3}{\left(y \cos{\left(y \right)} \right)} + 6 \left(y \sin{\left(y \right)} - \cos{\left(y \right)}\right) \left(y \cos{\left(y \right)} + 2 \sin{\left(y \right)}\right) \sin{\left(y \cos{\left(y \right)} \right)} \cos^{2}{\left(y \cos{\left(y \right)} \right)}\right)$$

Simplify

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Identical expressions

Derivative of 50sin(ycos(y))^(3)

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Derivative of 50*sin(y*cos(y))^(3)

The graph:

Piecewise:

The solution

Derivative of 50sin(ycos(y))^(3)