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4*cos(x)-tan(x)
The derivative of the function/ 4*cos(x)-tan(x)

Derivative of 4*cos(x)-tan(x)

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You have entered [src] 
4*cos(x) - tan(x)
4cos⁡(x)−tan⁡(x)4 \cos{\left(x \right)} - \tan{\left(x \right)} 
4*cos(x) - tan(x)
Detail solution

  1. Differentiate 4cos⁡(x)−tan⁡(x)4 \cos{\left(x \right)} - \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. The derivative of cosine is negative sine:

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

      So, the result is: −4sin⁡(x)- 4 \sin{\left(x \right)}

    2. 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: −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)cos⁡2(x)−4sin⁡(x)- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - 4 \sin{\left(x \right)}

  2. Now simplify:

    −sin⁡(x)+sin⁡(3x)+1cos⁡2(x)- \frac{\sin{\left(x \right)} + \sin{\left(3 x \right)} + 1}{\cos^{2}{\left(x \right)}}

The answer is:

−sin⁡(x)+sin⁡(3x)+1cos⁡2(x)- \frac{\sin{\left(x \right)} + \sin{\left(3 x \right)} + 1}{\cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-10001000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
        2              
-1 - tan (x) - 4*sin(x)
−4sin⁡(x)−tan⁡2(x)−1- 4 \sin{\left(x \right)} - \tan^{2}{\left(x \right)} - 1
Simplify
The second derivative [src] 
   /           /       2   \       \
-2*\2*cos(x) + \1 + tan (x)/*tan(x)/
−2((tan⁡2(x)+1)tan⁡(x)+2cos⁡(x))- 2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 \cos{\left(x \right)}\right)
Simplify
The third derivative [src] 
  /               2                                     \
  |  /       2   \                    2    /       2   \|
2*\- \1 + tan (x)/  + 2*sin(x) - 2*tan (x)*\1 + tan (x)//
2(−(tan⁡2(x)+1)2−2(tan⁡2(x)+1)tan⁡2(x)+2sin⁡(x))2 \left(- \left(\tan^{2}{\left(x \right)} + 1\right)^{2} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 2 \sin{\left(x \right)}\right)
Simplify
The graph
Derivative of 4*cos(x)-tan(x)
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