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

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

Function f() - derivative -N order at the point
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You have entered [src] 
sin(x)           
------ - 3*cot(x)
  3
sin⁡(x)3−3cot⁡(x)\frac{\sin{\left(x \right)}}{3} - 3 \cot{\left(x \right)} 
d /sin(x)           \
--|------ - 3*cot(x)|
dx\  3              /
ddx(sin⁡(x)3−3cot⁡(x))\frac{d}{d x} \left(\frac{\sin{\left(x \right)}}{3} - 3 \cot{\left(x \right)}\right)
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Detail solution

  1. Differentiate sin⁡(x)3−3cot⁡(x)\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:

        ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

      So, the result is: cos⁡(x)3\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

          1. Rewrite the function to be differentiated:

            cot⁡(x)=1tan⁡(x)\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}

          2. Let u=tan⁡(x)u = \tan{\left(x \right)}.

          3. Apply the power rule: 1u\frac{1}{u} goes to −1u2- \frac{1}{u^{2}}

          4. Then, apply the chain rule. Multiply by ddxtan⁡(x)\frac{d}{d x} \tan{\left(x \right)}:

            1. Rewrite the function to be differentiated:

              tan⁡(x)=sin⁡(x)cos⁡(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

            2. Apply the quotient rule, which is:

              ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} and g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}.

              To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

              1. The derivative of sine is cosine:

                ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

              To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

              1. The derivative of cosine is negative sine:

                ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

              Now plug in to the quotient rule:

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

            The result of the chain rule is:

            −sin⁡2(x)+cos⁡2(x)cos⁡2(x)tan⁡2(x)- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}

          Method #2

          1. Rewrite the function to be differentiated:

            cot⁡(x)=cos⁡(x)sin⁡(x)\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}

          2. Apply the quotient rule, which is:

            ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=cos⁡(x)f{\left(x \right)} = \cos{\left(x \right)} and g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}.

            To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

            1. The derivative of cosine is negative sine:

              ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

            To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

            1. The derivative of sine is cosine:

              ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

            Now plug in to the quotient rule:

            −sin⁡2(x)−cos⁡2(x)sin⁡2(x)\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}

        So, the result is: −3(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)- \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: 3(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)\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: 3(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)+cos⁡(x)3\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}

  2. Now simplify:

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

The answer is:

cos⁡(x)3+3sin⁡2(x)\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
cos⁡(x)3+3cot⁡2(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(cot⁡2(x)+1)cot⁡(x)+sin⁡(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(cot⁡2(x)+1)2+12(cot⁡2(x)+1)cot⁡2(x)−cos⁡(x)36 \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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