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The derivative of the function/ cosx*sinx*tgx

Derivative of cosx*sinx*tgx

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
cos(x)*sin(x)*tan(x)
sin⁡(x)cos⁡(x)tan⁡(x)\sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)} 
(cos(x)*sin(x))*tan(x)
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. 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)=cos⁡(x)f{\left(x \right)} = \cos{\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)}

      g(x)=sin⁡(x)g{\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)}

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

    g(x)=tan⁡(x)g{\left(x \right)} = \tan{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\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 is: (−sin⁡2(x)+cos⁡2(x))tan⁡(x)+(sin⁡2(x)+cos⁡2(x))sin⁡(x)cos⁡(x)\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)} + \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}}

  2. Now simplify:

    sin⁡(2x)\sin{\left(2 x \right)}

The answer is:

sin⁡(2x)\sin{\left(2 x \right)}

The graph
02468-8-6-4-2-10102-2
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The first derivative [src] 
/   2         2   \          /       2   \              
\cos (x) - sin (x)/*tan(x) + \1 + tan (x)/*cos(x)*sin(x)
(−sin⁡2(x)+cos⁡2(x))tan⁡(x)+(tan⁡2(x)+1)sin⁡(x)cos⁡(x)\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}
Simplify
The second derivative [src] 
  /  /       2   \ /   2         2   \                            /       2   \                     \
2*\- \1 + tan (x)/*\sin (x) - cos (x)/ - 2*cos(x)*sin(x)*tan(x) + \1 + tan (x)/*cos(x)*sin(x)*tan(x)/
2(−(sin⁡2(x)−cos⁡2(x))(tan⁡2(x)+1)+(tan⁡2(x)+1)sin⁡(x)cos⁡(x)tan⁡(x)−2sin⁡(x)cos⁡(x)tan⁡(x))2 \left(- \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)}\right)
Simplify
The third derivative [src] 
  /  /   2         2   \            /       2   \                   /       2   \ /   2         2   \          /       2   \ /         2   \              \
2*\2*\sin (x) - cos (x)/*tan(x) - 6*\1 + tan (x)/*cos(x)*sin(x) - 3*\1 + tan (x)/*\sin (x) - cos (x)/*tan(x) + \1 + tan (x)/*\1 + 3*tan (x)/*cos(x)*sin(x)/
2(−3(sin⁡2(x)−cos⁡2(x))(tan⁡2(x)+1)tan⁡(x)+2(sin⁡2(x)−cos⁡2(x))tan⁡(x)+(tan⁡2(x)+1)(3tan⁡2(x)+1)sin⁡(x)cos⁡(x)−6(tan⁡2(x)+1)sin⁡(x)cos⁡(x))2 \left(- 3 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} - 6 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}\right)
Simplify
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