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tan(5x^7+3x^4)
The derivative of the function/ tan(5x^7+3x^4)

Derivative of tan(5x^7+3x^4)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
   /   7      4\
tan\5*x  + 3*x /
tan⁡(5x7+3x4)\tan{\left(5 x^{7} + 3 x^{4} \right)} 
tan(5*x^7 + 3*x^4)
Detail solution

  1. Rewrite the function to be differentiated:

    tan⁡(5x7+3x4)=sin⁡(5x7+3x4)cos⁡(5x7+3x4)\tan{\left(5 x^{7} + 3 x^{4} \right)} = \frac{\sin{\left(5 x^{7} + 3 x^{4} \right)}}{\cos{\left(5 x^{7} + 3 x^{4} \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⁡(5x7+3x4)f{\left(x \right)} = \sin{\left(5 x^{7} + 3 x^{4} \right)} and g(x)=cos⁡(5x7+3x4)g{\left(x \right)} = \cos{\left(5 x^{7} + 3 x^{4} \right)}.

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

    1. Let u=5x7+3x4u = 5 x^{7} + 3 x^{4}.

    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 ddx(5x7+3x4)\frac{d}{d x} \left(5 x^{7} + 3 x^{4}\right):

      1. Differentiate 5x7+3x45 x^{7} + 3 x^{4} 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: x7x^{7} goes to 7x67 x^{6}

          So, the result is: 35x635 x^{6}

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

          1. Apply the power rule: x4x^{4} goes to 4x34 x^{3}

          So, the result is: 12x312 x^{3}

        The result is: 35x6+12x335 x^{6} + 12 x^{3}

      The result of the chain rule is:

      (35x6+12x3)cos⁡(5x7+3x4)\left(35 x^{6} + 12 x^{3}\right) \cos{\left(5 x^{7} + 3 x^{4} \right)}

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

    1. Let u=5x7+3x4u = 5 x^{7} + 3 x^{4}.

    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 ddx(5x7+3x4)\frac{d}{d x} \left(5 x^{7} + 3 x^{4}\right):

      1. Differentiate 5x7+3x45 x^{7} + 3 x^{4} 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: x7x^{7} goes to 7x67 x^{6}

          So, the result is: 35x635 x^{6}

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

          1. Apply the power rule: x4x^{4} goes to 4x34 x^{3}

          So, the result is: 12x312 x^{3}

        The result is: 35x6+12x335 x^{6} + 12 x^{3}

      The result of the chain rule is:

      −(35x6+12x3)sin⁡(5x7+3x4)- \left(35 x^{6} + 12 x^{3}\right) \sin{\left(5 x^{7} + 3 x^{4} \right)}

    Now plug in to the quotient rule:

    (35x6+12x3)sin⁡2(5x7+3x4)+(35x6+12x3)cos⁡2(5x7+3x4)cos⁡2(5x7+3x4)\frac{\left(35 x^{6} + 12 x^{3}\right) \sin^{2}{\left(5 x^{7} + 3 x^{4} \right)} + \left(35 x^{6} + 12 x^{3}\right) \cos^{2}{\left(5 x^{7} + 3 x^{4} \right)}}{\cos^{2}{\left(5 x^{7} + 3 x^{4} \right)}}

  3. Now simplify:

    x3(35x3+12)cos⁡2(x4(5x3+3))\frac{x^{3} \left(35 x^{3} + 12\right)}{\cos^{2}{\left(x^{4} \left(5 x^{3} + 3\right) \right)}}

The answer is:

x3(35x3+12)cos⁡2(x4(5x3+3))\frac{x^{3} \left(35 x^{3} + 12\right)}{\cos^{2}{\left(x^{4} \left(5 x^{3} + 3\right) \right)}}

The graph
02468-8-6-4-2-1010-2000000000020000000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
/       2/   7      4\\ /    3       6\
\1 + tan \5*x  + 3*x //*\12*x  + 35*x /
(35x6+12x3)(tan⁡2(5x7+3x4)+1)\left(35 x^{6} + 12 x^{3}\right) \left(\tan^{2}{\left(5 x^{7} + 3 x^{4} \right)} + 1\right)
Simplify
The second derivative [src] 
                               /                             2                   \
   2 /       2/ 4 /       3\\\ |          3    4 /         3\     / 4 /       3\\|
2*x *\1 + tan \x *\3 + 5*x ///*\18 + 105*x  + x *\12 + 35*x / *tan\x *\3 + 5*x ///
2x2(tan⁡2(x4(5x3+3))+1)(x4(35x3+12)2tan⁡(x4(5x3+3))+105x3+18)2 x^{2} \left(\tan^{2}{\left(x^{4} \left(5 x^{3} + 3\right) \right)} + 1\right) \left(x^{4} \left(35 x^{3} + 12\right)^{2} \tan{\left(x^{4} \left(5 x^{3} + 3\right) \right)} + 105 x^{3} + 18\right)
Simplify
The third derivative [src] 
                              /                             3                                              3                                                                        \
    /       2/ 4 /       3\\\ |          3    8 /         3\  /       2/ 4 /       3\\\      8 /         3\     2/ 4 /       3\\       4 /        3\ /         3\    / 4 /       3\\|
2*x*\1 + tan \x *\3 + 5*x ///*\36 + 525*x  + x *\12 + 35*x / *\1 + tan \x *\3 + 5*x /// + 2*x *\12 + 35*x / *tan \x *\3 + 5*x // + 18*x *\6 + 35*x /*\12 + 35*x /*tan\x *\3 + 5*x ///
2x(tan⁡2(x4(5x3+3))+1)(x8(35x3+12)3(tan⁡2(x4(5x3+3))+1)+2x8(35x3+12)3tan⁡2(x4(5x3+3))+18x4(35x3+6)(35x3+12)tan⁡(x4(5x3+3))+525x3+36)2 x \left(\tan^{2}{\left(x^{4} \left(5 x^{3} + 3\right) \right)} + 1\right) \left(x^{8} \left(35 x^{3} + 12\right)^{3} \left(\tan^{2}{\left(x^{4} \left(5 x^{3} + 3\right) \right)} + 1\right) + 2 x^{8} \left(35 x^{3} + 12\right)^{3} \tan^{2}{\left(x^{4} \left(5 x^{3} + 3\right) \right)} + 18 x^{4} \left(35 x^{3} + 6\right) \left(35 x^{3} + 12\right) \tan{\left(x^{4} \left(5 x^{3} + 3\right) \right)} + 525 x^{3} + 36\right)
Simplify
The graph
Derivative of tan(5x^7+3x^4)
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