Mister Exam

Derivative of cosecxtanx

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

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cos(e)*c*x*tan(x)
cxcos(e)tan(x)c x \cos{\left(e \right)} \tan{\left(x \right)}
d                    
--(cos(e)*c*x*tan(x))
dx                   
xcxcos(e)tan(x)\frac{\partial}{\partial x} c x \cos{\left(e \right)} \tan{\left(x \right)}
Detail solution
  1. The derivative of a constant times a function is the constant times the derivative of the function.

    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)=xf{\left(x \right)} = x; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Apply the power rule: xx goes to 11

      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:

        sin2(x)+cos2(x)cos2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

      The result is: x(sin2(x)+cos2(x))cos2(x)+tan(x)\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}

    So, the result is: c(x(sin2(x)+cos2(x))cos2(x)+tan(x))cos(e)c \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}\right) \cos{\left(e \right)}

  2. Now simplify:

    c(xcos2(x)+tan(x))cos(e)c \left(\frac{x}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}\right) \cos{\left(e \right)}


The answer is:

c(xcos2(x)+tan(x))cos(e)c \left(\frac{x}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}\right) \cos{\left(e \right)}

The first derivative [src]
                      /       2   \       
c*cos(e)*tan(x) + c*x*\1 + tan (x)/*cos(e)
cx(tan2(x)+1)cos(e)+ccos(e)tan(x)c x \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(e \right)} + c \cos{\left(e \right)} \tan{\left(x \right)}
The second derivative [src]
    /       2   \                      
2*c*\1 + tan (x)/*(1 + x*tan(x))*cos(e)
2c(xtan(x)+1)(tan2(x)+1)cos(e)2 c \left(x \tan{\left(x \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(e \right)}
The third derivative [src]
    /       2   \ /             /         2   \\       
2*c*\1 + tan (x)/*\3*tan(x) + x*\1 + 3*tan (x)//*cos(e)
2c(x(3tan2(x)+1)+3tan(x))(tan2(x)+1)cos(e)2 c \left(x \left(3 \tan^{2}{\left(x \right)} + 1\right) + 3 \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(e \right)}