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

Derivative of tan(x)^3+sin(3x^2)

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

You have entered [src] 
   3         /   2\
tan (x) + sin\3*x /
sin⁡(3x2)+tan⁡3(x)\sin{\left(3 x^{2} \right)} + \tan^{3}{\left(x \right)} 
d /   3         /   2\\
--\tan (x) + sin\3*x //
dx
ddx(sin⁡(3x2)+tan⁡3(x))\frac{d}{d x} \left(\sin{\left(3 x^{2} \right)} + \tan^{3}{\left(x \right)}\right)
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Detail solution

  1. Differentiate sin⁡(3x2)+tan⁡3(x)\sin{\left(3 x^{2} \right)} + \tan^{3}{\left(x \right)} term by term:

    1. Let u=tan⁡(x)u = \tan{\left(x \right)}.

    2. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

    3. Then, apply the chain rule. Multiply by ddxtan⁡(x)\frac{d}{d x} \tan{\left(x \right)}:

      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)}}

      The result of the chain rule is:

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

    4. Let u=3x2u = 3 x^{2}.

    5. The derivative of sine is cosine:

      ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

    6. Then, apply the chain rule. Multiply by ddx3x2\frac{d}{d x} 3 x^{2}:

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

        1. Apply the power rule: x2x^{2} goes to 2x2 x

        So, the result is: 6x6 x

      The result of the chain rule is:

      6xcos⁡(3x2)6 x \cos{\left(3 x^{2} \right)}

    The result is: 6xcos⁡(3x2)+3(sin⁡2(x)+cos⁡2(x))tan⁡2(x)cos⁡2(x)6 x \cos{\left(3 x^{2} \right)} + \frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

  2. Now simplify:

    6xcos⁡(3x2)+3tan⁡2(x)cos⁡2(x)6 x \cos{\left(3 x^{2} \right)} + \frac{3 \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

The answer is:

6xcos⁡(3x2)+3tan⁡2(x)cos⁡2(x)6 x \cos{\left(3 x^{2} \right)} + \frac{3 \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-10104000000-2000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   2    /         2   \          /   2\
tan (x)*\3 + 3*tan (x)/ + 6*x*cos\3*x /
6xcos⁡(3x2)+(3tan⁡2(x)+3)tan⁡2(x)6 x \cos{\left(3 x^{2} \right)} + \left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)}
Simplify
The second derivative [src] 
  /             2                                                            \
  |/       2   \              3    /       2   \      2    /   2\      /   2\|
6*\\1 + tan (x)/ *tan(x) + tan (x)*\1 + tan (x)/ - 6*x *sin\3*x / + cos\3*x //
6(−6x2sin⁡(3x2)+(tan⁡2(x)+1)2tan⁡(x)+(tan⁡2(x)+1)tan⁡3(x)+cos⁡(3x2))6 \left(- 6 x^{2} \sin{\left(3 x^{2} \right)} + \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{3}{\left(x \right)} + \cos{\left(3 x^{2} \right)}\right)
Simplify
The third derivative [src] 
  /             3                                                                               2        \
  |/       2   \        3    /   2\           /   2\        4    /       2   \     /       2   \     2   |
6*\\1 + tan (x)/  - 36*x *cos\3*x / - 18*x*sin\3*x / + 2*tan (x)*\1 + tan (x)/ + 7*\1 + tan (x)/ *tan (x)/
6(−36x3cos⁡(3x2)−18xsin⁡(3x2)+(tan⁡2(x)+1)3+7(tan⁡2(x)+1)2tan⁡2(x)+2(tan⁡2(x)+1)tan⁡4(x))6 \left(- 36 x^{3} \cos{\left(3 x^{2} \right)} - 18 x \sin{\left(3 x^{2} \right)} + \left(\tan^{2}{\left(x \right)} + 1\right)^{3} + 7 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan^{2}{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{4}{\left(x \right)}\right)
Simplify
The graph
Derivative of tan(x)^3+sin(3x^2)
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