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The derivative of the function/ 2*tan(x)-4*x+pi+13

Derivative of 2*tan(x)-4*x+pi+13

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You have entered [src] 
2*tan(x) - 4*x + pi + 13
((−4x+2tan⁡(x))+π)+13\left(\left(- 4 x + 2 \tan{\left(x \right)}\right) + \pi\right) + 13 
2*tan(x) - 4*x + pi + 13
Detail solution

  1. Differentiate ((−4x+2tan⁡(x))+π)+13\left(\left(- 4 x + 2 \tan{\left(x \right)}\right) + \pi\right) + 13 term by term:

    1. Differentiate (−4x+2tan⁡(x))+π\left(- 4 x + 2 \tan{\left(x \right)}\right) + \pi term by term:

      1. Differentiate −4x+2tan⁡(x)- 4 x + 2 \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: 2(sin⁡2(x)+cos⁡2(x))cos⁡2(x)\frac{2 \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. Apply the power rule: xx goes to 11

          So, the result is: −4-4

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

      2. The derivative of the constant π\pi is zero.

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

    2. The derivative of the constant 1313 is zero.

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

  2. Now simplify:

    2tan⁡2(x)−22 \tan^{2}{\left(x \right)} - 2

The answer is:

2tan⁡2(x)−22 \tan^{2}{\left(x \right)} - 2

The graph
02468-8-6-4-2-1010-20002000
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The first derivative [src] 
          2   
-2 + 2*tan (x)
2tan⁡2(x)−22 \tan^{2}{\left(x \right)} - 2
Simplify
The second derivative [src] 
  /       2   \       
4*\1 + tan (x)/*tan(x)
4(tan⁡2(x)+1)tan⁡(x)4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}
Simplify
The third derivative [src] 
  /       2   \ /         2   \
4*\1 + tan (x)/*\1 + 3*tan (x)/
4(tan⁡2(x)+1)(3tan⁡2(x)+1)4 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right)
Simplify
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