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The derivative of the function/ 2*tan(3x)+3*tan(2x)

Derivative of 2*tan(3x)+3*tan(2x)

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2*tan(3*x) + 3*tan(2*x)
3tan⁡(2x)+2tan⁡(3x)3 \tan{\left(2 x \right)} + 2 \tan{\left(3 x \right)} 
2*tan(3*x) + 3*tan(2*x)
Detail solution

  1. Differentiate 3tan⁡(2x)+2tan⁡(3x)3 \tan{\left(2 x \right)} + 2 \tan{\left(3 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⁡(3x)=sin⁡(3x)cos⁡(3x)\tan{\left(3 x \right)} = \frac{\sin{\left(3 x \right)}}{\cos{\left(3 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⁡(3x)f{\left(x \right)} = \sin{\left(3 x \right)} and g(x)=cos⁡(3x)g{\left(x \right)} = \cos{\left(3 x \right)}.

        To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. Let u=3xu = 3 x.

        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 ddx3x\frac{d}{d x} 3 x:

          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: 33

          The result of the chain rule is:

          3cos⁡(3x)3 \cos{\left(3 x \right)}

        To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. Let u=3xu = 3 x.

        2. The derivative of cosine is negative sine:

          dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

        3. Then, apply the chain rule. Multiply by ddx3x\frac{d}{d x} 3 x:

          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: 33

          The result of the chain rule is:

          −3sin⁡(3x)- 3 \sin{\left(3 x \right)}

        Now plug in to the quotient rule:

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

      So, the result is: 2(3sin⁡2(3x)+3cos⁡2(3x))cos⁡2(3x)\frac{2 \left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right)}{\cos^{2}{\left(3 x \right)}}

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

      1. Let u=2xu = 2 x.

      2. ddutan⁡(u)=1cos⁡2(u)\frac{d}{d u} \tan{\left(u \right)} = \frac{1}{\cos^{2}{\left(u \right)}}

      3. Then, apply the chain rule. Multiply by ddx2x\frac{d}{d x} 2 x:

        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

        The result of the chain rule is:

        2cos⁡2(2x)\frac{2}{\cos^{2}{\left(2 x \right)}}

      So, the result is: 3(2sin⁡2(2x)+2cos⁡2(2x))cos⁡2(2x)\frac{3 \left(2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right)}{\cos^{2}{\left(2 x \right)}}

    The result is: 3(2sin⁡2(2x)+2cos⁡2(2x))cos⁡2(2x)+2(3sin⁡2(3x)+3cos⁡2(3x))cos⁡2(3x)\frac{3 \left(2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right)}{\cos^{2}{\left(2 x \right)}} + \frac{2 \left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right)}{\cos^{2}{\left(3 x \right)}}

  2. Now simplify:

    6cos⁡2(3x)+6cos⁡2(2x)\frac{6}{\cos^{2}{\left(3 x \right)}} + \frac{6}{\cos^{2}{\left(2 x \right)}}

The answer is:

6cos⁡2(3x)+6cos⁡2(2x)\frac{6}{\cos^{2}{\left(3 x \right)}} + \frac{6}{\cos^{2}{\left(2 x \right)}}

The graph
02468-8-6-4-2-1010-200000200000
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The first derivative [src] 
          2             2     
12 + 6*tan (2*x) + 6*tan (3*x)
6tan⁡2(2x)+6tan⁡2(3x)+126 \tan^{2}{\left(2 x \right)} + 6 \tan^{2}{\left(3 x \right)} + 12
Simplify
The second derivative [src] 
   /  /       2     \              /       2     \         \
12*\2*\1 + tan (2*x)/*tan(2*x) + 3*\1 + tan (3*x)/*tan(3*x)/
12(2(tan⁡2(2x)+1)tan⁡(2x)+3(tan⁡2(3x)+1)tan⁡(3x))12 \left(2 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)} + 3 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)}\right)
Simplify
3-я производная [src] 
   /                 2                    2                                                             \
   |  /       2     \      /       2     \         2      /       2     \         2      /       2     \|
12*\4*\1 + tan (2*x)/  + 9*\1 + tan (3*x)/  + 8*tan (2*x)*\1 + tan (2*x)/ + 18*tan (3*x)*\1 + tan (3*x)//
12(4(tan⁡2(2x)+1)2+8(tan⁡2(2x)+1)tan⁡2(2x)+9(tan⁡2(3x)+1)2+18(tan⁡2(3x)+1)tan⁡2(3x))12 \left(4 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2} + 8 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan^{2}{\left(2 x \right)} + 9 \left(\tan^{2}{\left(3 x \right)} + 1\right)^{2} + 18 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan^{2}{\left(3 x \right)}\right)
Simplify
The third derivative [src] 
   /                 2                    2                                                             \
   |  /       2     \      /       2     \         2      /       2     \         2      /       2     \|
12*\4*\1 + tan (2*x)/  + 9*\1 + tan (3*x)/  + 8*tan (2*x)*\1 + tan (2*x)/ + 18*tan (3*x)*\1 + tan (3*x)//
12(4(tan⁡2(2x)+1)2+8(tan⁡2(2x)+1)tan⁡2(2x)+9(tan⁡2(3x)+1)2+18(tan⁡2(3x)+1)tan⁡2(3x))12 \left(4 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2} + 8 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan^{2}{\left(2 x \right)} + 9 \left(\tan^{2}{\left(3 x \right)} + 1\right)^{2} + 18 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan^{2}{\left(3 x \right)}\right)
Simplify
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