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3*sin(tan(5*x+pi))
The derivative of the function/ 3*sin(tan(5*x+pi))

Derivative of 3*sin(tan(5*x+pi))

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The solution

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3*sin(tan(5*x + pi))
3sin⁡(tan⁡(5x+π))3 \sin{\left(\tan{\left(5 x + \pi \right)} \right)} 
3*sin(tan(5*x + pi))
Detail solution

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

    1. Let u=tan⁡(5x+π)u = \tan{\left(5 x + \pi \right)}.

    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 ddxtan⁡(5x+π)\frac{d}{d x} \tan{\left(5 x + \pi \right)}:

      1. Rewrite the function to be differentiated:

        tan⁡(5x+π)=sin⁡(5x)cos⁡(5x)\tan{\left(5 x + \pi \right)} = \frac{\sin{\left(5 x \right)}}{\cos{\left(5 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⁡(5x)f{\left(x \right)} = \sin{\left(5 x \right)} and g(x)=cos⁡(5x)g{\left(x \right)} = \cos{\left(5 x \right)}.

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

        1. Let u=5xu = 5 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 ddx5x\frac{d}{d x} 5 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: 55

          The result of the chain rule is:

          5cos⁡(5x)5 \cos{\left(5 x \right)}

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

        1. Let u=5xu = 5 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 ddx5x\frac{d}{d x} 5 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: 55

          The result of the chain rule is:

          −5sin⁡(5x)- 5 \sin{\left(5 x \right)}

        Now plug in to the quotient rule:

        5sin⁡2(5x)+5cos⁡2(5x)cos⁡2(5x)\frac{5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}}

      The result of the chain rule is:

      (5sin⁡2(5x)+5cos⁡2(5x))cos⁡(tan⁡(5x+π))cos⁡2(5x)\frac{\left(5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}\right) \cos{\left(\tan{\left(5 x + \pi \right)} \right)}}{\cos^{2}{\left(5 x \right)}}

    So, the result is: 3(5sin⁡2(5x)+5cos⁡2(5x))cos⁡(tan⁡(5x+π))cos⁡2(5x)\frac{3 \left(5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}\right) \cos{\left(\tan{\left(5 x + \pi \right)} \right)}}{\cos^{2}{\left(5 x \right)}}

  2. Now simplify:

    15cos⁡(tan⁡(5x))cos⁡2(5x)\frac{15 \cos{\left(\tan{\left(5 x \right)} \right)}}{\cos^{2}{\left(5 x \right)}}

The answer is:

15cos⁡(tan⁡(5x))cos⁡2(5x)\frac{15 \cos{\left(\tan{\left(5 x \right)} \right)}}{\cos^{2}{\left(5 x \right)}}

The graph
02468-8-6-4-2-1010-1000010000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  /         2          \                   
3*\5 + 5*tan (5*x + pi)/*cos(tan(5*x + pi))
3(5tan⁡2(5x+π)+5)cos⁡(tan⁡(5x+π))3 \left(5 \tan^{2}{\left(5 x + \pi \right)} + 5\right) \cos{\left(\tan{\left(5 x + \pi \right)} \right)}
Simplify
The second derivative [src] 
    /       2     \ //       2     \                                         \
-75*\1 + tan (5*x)/*\\1 + tan (5*x)/*sin(tan(5*x)) - 2*cos(tan(5*x))*tan(5*x)/
−75((tan⁡2(5x)+1)sin⁡(tan⁡(5x))−2cos⁡(tan⁡(5x))tan⁡(5x))(tan⁡2(5x)+1)- 75 \left(\left(\tan^{2}{\left(5 x \right)} + 1\right) \sin{\left(\tan{\left(5 x \right)} \right)} - 2 \cos{\left(\tan{\left(5 x \right)} \right)} \tan{\left(5 x \right)}\right) \left(\tan^{2}{\left(5 x \right)} + 1\right)
Simplify
The third derivative [src] 
                     /               2                                                                                                                       \
     /       2     \ |/       2     \                       2                        /       2     \                   /       2     \                       |
-375*\1 + tan (5*x)/*\\1 + tan (5*x)/ *cos(tan(5*x)) - 4*tan (5*x)*cos(tan(5*x)) - 2*\1 + tan (5*x)/*cos(tan(5*x)) + 6*\1 + tan (5*x)/*sin(tan(5*x))*tan(5*x)/
−375(tan⁡2(5x)+1)((tan⁡2(5x)+1)2cos⁡(tan⁡(5x))+6(tan⁡2(5x)+1)sin⁡(tan⁡(5x))tan⁡(5x)−2(tan⁡2(5x)+1)cos⁡(tan⁡(5x))−4cos⁡(tan⁡(5x))tan⁡2(5x))- 375 \left(\tan^{2}{\left(5 x \right)} + 1\right) \left(\left(\tan^{2}{\left(5 x \right)} + 1\right)^{2} \cos{\left(\tan{\left(5 x \right)} \right)} + 6 \left(\tan^{2}{\left(5 x \right)} + 1\right) \sin{\left(\tan{\left(5 x \right)} \right)} \tan{\left(5 x \right)} - 2 \left(\tan^{2}{\left(5 x \right)} + 1\right) \cos{\left(\tan{\left(5 x \right)} \right)} - 4 \cos{\left(\tan{\left(5 x \right)} \right)} \tan^{2}{\left(5 x \right)}\right)
Simplify
The graph
Derivative of 3*sin(tan(5*x+pi))
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