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y=tgx*sin(2x-5)

Derivative of y=tgx*sin(2x-5)

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

You have entered [src]
tan(x)*sin(2*x - 5)
sin(2x5)tan(x)\sin{\left(2 x - 5 \right)} \tan{\left(x \right)}
tan(x)*sin(2*x - 5)
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)=tan(x)f{\left(x \right)} = \tan{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\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)}}

    g(x)=sin(2x5)g{\left(x \right)} = \sin{\left(2 x - 5 \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=2x5u = 2 x - 5.

    2. The derivative of sine is cosine:

      ddusin(u)=cos(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

    3. Then, apply the chain rule. Multiply by ddx(2x5)\frac{d}{d x} \left(2 x - 5\right):

      1. Differentiate 2x52 x - 5 term by term:

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: xx goes to 11

          So, the result is: 22

        2. The derivative of the constant 5-5 is zero.

        The result is: 22

      The result of the chain rule is:

      2cos(2x5)2 \cos{\left(2 x - 5 \right)}

    The result is: (sin2(x)+cos2(x))sin(2x5)cos2(x)+2cos(2x5)tan(x)\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(2 x - 5 \right)}}{\cos^{2}{\left(x \right)}} + 2 \cos{\left(2 x - 5 \right)} \tan{\left(x \right)}

  2. Now simplify:

    sin(2x5)+sin(4x5)2+sin(5)2cos2(x)\frac{\sin{\left(2 x - 5 \right)} + \frac{\sin{\left(4 x - 5 \right)}}{2} + \frac{\sin{\left(5 \right)}}{2}}{\cos^{2}{\left(x \right)}}


The answer is:

sin(2x5)+sin(4x5)2+sin(5)2cos2(x)\frac{\sin{\left(2 x - 5 \right)} + \frac{\sin{\left(4 x - 5 \right)}}{2} + \frac{\sin{\left(5 \right)}}{2}}{\cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-10001000
The first derivative [src]
/       2   \                                     
\1 + tan (x)/*sin(2*x - 5) + 2*cos(2*x - 5)*tan(x)
(tan2(x)+1)sin(2x5)+2cos(2x5)tan(x)\left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(2 x - 5 \right)} + 2 \cos{\left(2 x - 5 \right)} \tan{\left(x \right)}
The second derivative [src]
  /                            /       2   \                 /       2   \                     \
2*\-2*sin(-5 + 2*x)*tan(x) + 2*\1 + tan (x)/*cos(-5 + 2*x) + \1 + tan (x)/*sin(-5 + 2*x)*tan(x)/
2((tan2(x)+1)sin(2x5)tan(x)+2(tan2(x)+1)cos(2x5)2sin(2x5)tan(x))2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(2 x - 5 \right)} \tan{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(2 x - 5 \right)} - 2 \sin{\left(2 x - 5 \right)} \tan{\left(x \right)}\right)
The third derivative [src]
  /    /       2   \                                          /       2   \ /         2   \                   /       2   \                     \
2*\- 6*\1 + tan (x)/*sin(-5 + 2*x) - 4*cos(-5 + 2*x)*tan(x) + \1 + tan (x)/*\1 + 3*tan (x)/*sin(-5 + 2*x) + 6*\1 + tan (x)/*cos(-5 + 2*x)*tan(x)/
2((tan2(x)+1)(3tan2(x)+1)sin(2x5)6(tan2(x)+1)sin(2x5)+6(tan2(x)+1)cos(2x5)tan(x)4cos(2x5)tan(x))2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \sin{\left(2 x - 5 \right)} - 6 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(2 x - 5 \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(2 x - 5 \right)} \tan{\left(x \right)} - 4 \cos{\left(2 x - 5 \right)} \tan{\left(x \right)}\right)
The graph
Derivative of y=tgx*sin(2x-5)