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The derivative of the function/ y=5*tgx-8*sinx

Derivative of y=5*tgx-8*sinx

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You have entered [src] 
5*tan(x) - 8*sin(x)
−8sin⁡(x)+5tan⁡(x)- 8 \sin{\left(x \right)} + 5 \tan{\left(x \right)} 
5*tan(x) - 8*sin(x)
Detail solution

  1. Differentiate −8sin⁡(x)+5tan⁡(x)- 8 \sin{\left(x \right)} + 5 \tan{\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. 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)}}

      So, the result is: 5(sin⁡2(x)+cos⁡2(x))cos⁡2(x)\frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}

    2. 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: −8cos⁡(x)- 8 \cos{\left(x \right)}

    The result is: 5(sin⁡2(x)+cos⁡2(x))cos⁡2(x)−8cos⁡(x)\frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - 8 \cos{\left(x \right)}

  2. Now simplify:

    −8cos⁡(x)+5cos⁡2(x)- 8 \cos{\left(x \right)} + \frac{5}{\cos^{2}{\left(x \right)}}

The answer is:

−8cos⁡(x)+5cos⁡2(x)- 8 \cos{\left(x \right)} + \frac{5}{\cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-50005000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                    2   
5 - 8*cos(x) + 5*tan (x)
−8cos⁡(x)+5tan⁡2(x)+5- 8 \cos{\left(x \right)} + 5 \tan^{2}{\left(x \right)} + 5
Simplify
The second derivative [src] 
  /             /       2   \       \
2*\4*sin(x) + 5*\1 + tan (x)/*tan(x)/
2(5(tan⁡2(x)+1)tan⁡(x)+4sin⁡(x))2 \left(5 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 4 \sin{\left(x \right)}\right)
Simplify
The third derivative [src] 
  /                          2                           \
  |             /       2   \          2    /       2   \|
2*\4*cos(x) + 5*\1 + tan (x)/  + 10*tan (x)*\1 + tan (x)//
2(5(tan⁡2(x)+1)2+10(tan⁡2(x)+1)tan⁡2(x)+4cos⁡(x))2 \left(5 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 10 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 4 \cos{\left(x \right)}\right)
Simplify
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