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(sinx+cosx)(tanx+cotx)
The derivative of the function/ (sinx+cosx)(tanx+cotx)

Derivative of (sinx+cosx)(tanx+cotx)

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(sin(x) + cos(x))*(tan(x) + cot(x))
(sin⁡(x)+cos⁡(x))(tan⁡(x)+cot⁡(x))\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) 
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--((sin(x) + cos(x))*(tan(x) + cot(x)))
dx
ddx(sin⁡(x)+cos⁡(x))(tan⁡(x)+cot⁡(x))\frac{d}{d x} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right)
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Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=sin⁡(x)+cos⁡(x)f{\left(x \right)} = \sin{\left(x \right)} + \cos{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Differentiate sin⁡(x)+cos⁡(x)\sin{\left(x \right)} + \cos{\left(x \right)} term by term:

      1. The derivative of sine is cosine:

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

      2. The derivative of cosine is negative sine:

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

      The result is: −sin⁡(x)+cos⁡(x)- \sin{\left(x \right)} + \cos{\left(x \right)}

    g(x)=tan⁡(x)+cot⁡(x)g{\left(x \right)} = \tan{\left(x \right)} + \cot{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate tan⁡(x)+cot⁡(x)\tan{\left(x \right)} + \cot{\left(x \right)} term by term:

      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)}}

      3. 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. ddxtan⁡(x)=1cos⁡2(x)\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\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)}}

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

    The result is: (sin⁡2(x)+cos⁡2(x)cos⁡2(x)−sin⁡2(x)+cos⁡2(x)cos⁡2(x)tan⁡2(x))(sin⁡(x)+cos⁡(x))+(−sin⁡(x)+cos⁡(x))(tan⁡(x)+cot⁡(x))\left(\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) + \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right)

  2. Now simplify:

    1cos⁡(x)+1sin⁡(x)+4sin⁡(x)cos⁡(2x)+1−82sin⁡(x+π4)1−cos⁡(4x)\frac{1}{\cos{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}} + \frac{4 \sin{\left(x \right)}}{\cos{\left(2 x \right)} + 1} - \frac{8 \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}}{1 - \cos{\left(4 x \right)}}

The answer is:

1cos⁡(x)+1sin⁡(x)+4sin⁡(x)cos⁡(2x)+1−82sin⁡(x+π4)1−cos⁡(4x)\frac{1}{\cos{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}} + \frac{4 \sin{\left(x \right)}}{\cos{\left(2 x \right)} + 1} - \frac{8 \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}}{1 - \cos{\left(4 x \right)}}

The graph
02468-8-6-4-2-1010-1000010000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
/   2         2   \                                                         
\tan (x) - cot (x)/*(sin(x) + cos(x)) + (-sin(x) + cos(x))*(tan(x) + cot(x))
(−sin⁡(x)+cos⁡(x))(tan⁡(x)+cot⁡(x))+(sin⁡(x)+cos⁡(x))(tan⁡2(x)−cot⁡2(x))\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) + \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right)
Simplify
The second derivative [src] 
                                         /   2         2   \                        //       2   \          /       2   \       \                  
-(cos(x) + sin(x))*(cot(x) + tan(x)) - 2*\tan (x) - cot (x)/*(-cos(x) + sin(x)) + 2*\\1 + cot (x)/*cot(x) + \1 + tan (x)/*tan(x)/*(cos(x) + sin(x))
2((tan⁡2(x)+1)tan⁡(x)+(cot⁡2(x)+1)cot⁡(x))(sin⁡(x)+cos⁡(x))−2(sin⁡(x)−cos⁡(x))(tan⁡2(x)−cot⁡2(x))−(sin⁡(x)+cos⁡(x))(tan⁡(x)+cot⁡(x))2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) - 2 \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right) - \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right)
Simplify
The third derivative [src] 
                                                                                                                                                                          /             2                2                                                    \
                                                            //       2   \          /       2   \       \     /   2         2   \                                         |/       2   \    /       2   \         2    /       2   \        2    /       2   \|
(-cos(x) + sin(x))*(cot(x) + tan(x)) - 6*(-cos(x) + sin(x))*\\1 + cot (x)/*cot(x) + \1 + tan (x)/*tan(x)/ - 3*\tan (x) - cot (x)/*(cos(x) + sin(x)) + 2*(cos(x) + sin(x))*\\1 + tan (x)/  - \1 + cot (x)/  - 2*cot (x)*\1 + cot (x)/ + 2*tan (x)*\1 + tan (x)//
−6((tan⁡2(x)+1)tan⁡(x)+(cot⁡2(x)+1)cot⁡(x))(sin⁡(x)−cos⁡(x))+(sin⁡(x)−cos⁡(x))(tan⁡(x)+cot⁡(x))−3(sin⁡(x)+cos⁡(x))(tan⁡2(x)−cot⁡2(x))+2(sin⁡(x)+cos⁡(x))((tan⁡2(x)+1)2+2(tan⁡2(x)+1)tan⁡2(x)−(cot⁡2(x)+1)2−2(cot⁡2(x)+1)cot⁡2(x))- 6 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right) \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) + \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) - 3 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right) + 2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} - \left(\cot^{2}{\left(x \right)} + 1\right)^{2} - 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)}\right)
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The graph
Derivative of (sinx+cosx)(tanx+cotx)
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