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13*x-13*tan(x)-18
The derivative of the function/ 13*x-13*tan(x)-18

Derivative of 13*x-13*tan(x)-18

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You have entered [src] 
13*x - 13*tan(x) - 18
(13x−13tan⁡(x))−18\left(13 x - 13 \tan{\left(x \right)}\right) - 18 
13*x - 13*tan(x) - 18
Detail solution

  1. Differentiate (13x−13tan⁡(x))−18\left(13 x - 13 \tan{\left(x \right)}\right) - 18 term by term:

    1. Differentiate 13x−13tan⁡(x)13 x - 13 \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. Apply the power rule: xx goes to 11

        So, the result is: 1313

      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: −13(sin⁡2(x)+cos⁡2(x))cos⁡2(x)- \frac{13 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}

      The result is: −13(sin⁡2(x)+cos⁡2(x))cos⁡2(x)+13- \frac{13 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + 13

    2. The derivative of the constant −18-18 is zero.

    The result is: −13(sin⁡2(x)+cos⁡2(x))cos⁡2(x)+13- \frac{13 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + 13

  2. Now simplify:

    −13tan⁡2(x)- 13 \tan^{2}{\left(x \right)}

The answer is:

−13tan⁡2(x)- 13 \tan^{2}{\left(x \right)}

The graph
02468-8-6-4-2-1010-2000010000
Plot the graph f(x)
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The first derivative [src] 
       2   
-13*tan (x)
−13tan⁡2(x)- 13 \tan^{2}{\left(x \right)}
Simplify
The second derivative [src] 
    /       2   \       
-26*\1 + tan (x)/*tan(x)
−26(tan⁡2(x)+1)tan⁡(x)- 26 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}
Simplify
The third derivative [src] 
    /       2   \ /         2   \
-26*\1 + tan (x)/*\1 + 3*tan (x)/
−26(tan⁡2(x)+1)(3tan⁡2(x)+1)- 26 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right)
Simplify
The graph
Derivative of 13*x-13*tan(x)-18
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