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Identical expressions
y= one / three *sinx-3ctgx
y equally 1 divide by 3 multiply by sinus of x minus 3ctgx
y equally one divide by three multiply by sinus of x minus 3ctgx
y=1/3sinx-3ctgx
y=1 divide by 3*sinx-3ctgx
Similar expressions
y=1/3*sinx+3ctgx

The derivative of the function/ y=1/3*sinx-3ctgx

Derivative of y=1/3*sinx-3ctgx

The solution

You have entered [src]

sin(x)           
------ - 3*cot(x)
  3

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

d /sin(x)           \
--|------ - 3*cot(x)|
dx\  3              /

$$\frac{d}{d x} \left(\frac{\sin{\left(x \right)}}{3} - 3 \cot{\left(x \right)}\right)$$

Transform

Detail solution

Differentiate $\frac{\sin{\left(x \right)}}{3} - 3 \cot{\left(x \right)}$ 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 sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  So, the result is: $\frac{\cos{\left(x \right)}}{3}$
2. 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. There are multiple ways to do this derivative.
      Method #1
      
      Rewrite the function to be differentiated:
      
      $\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}$
      
      Let $u = \tan{\left(x \right)}$ .
      
      Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
      
      Rewrite the function to be differentiated:
      
      $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
      
      Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      
      Now plug in to the quotient rule:
      
      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
      
      The result of the chain rule is:
      
      $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
      Method #2
      
      Rewrite the function to be differentiated:
      
      $\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
      
      Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(x \right)}$ and $g{\left(x \right)} = \sin{\left(x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      
      Now plug in to the quotient rule:
      
      $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
    So, the result is: $- \frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
  So, the result is: $\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
The result is: $\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \frac{\cos{\left(x \right)}}{3}$
Now simplify:

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

The answer is:

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

The first derivative [src]

         2      cos(x)
3 + 3*cot (x) + ------
                  3

$$\frac{\cos{\left(x \right)}}{3} + 3 \cot^{2}{\left(x \right)} + 3$$

Simplify

The second derivative [src]

 /sin(x)     /       2   \       \
-|------ + 6*\1 + cot (x)/*cot(x)|
 \  3                            /

$$- (6 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \frac{\sin{\left(x \right)}}{3})$$

Simplify

The third derivative [src]

               2                                    
  /       2   \    cos(x)         2    /       2   \
6*\1 + cot (x)/  - ------ + 12*cot (x)*\1 + cot (x)/
                     3

$$6 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} + 12 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} - \frac{\cos{\left(x \right)}}{3}$$

Simplify

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Derivative of y=1/3*sinx-3ctgx

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The solution

Method #1

Method #2