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sin^8(3x^3-5x)
The derivative of the function/ sin^8(3x^3-5x)

Derivative of sin^8(3x^3-5x)

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The solution

You have entered [src] 
   8/   3      \
sin \3*x  - 5*x/
sin⁡8(3x3−5x)\sin^{8}{\left(3 x^{3} - 5 x \right)} 
d /   8/   3      \\
--\sin \3*x  - 5*x//
dx
ddxsin⁡8(3x3−5x)\frac{d}{d x} \sin^{8}{\left(3 x^{3} - 5 x \right)}
Transform
Detail solution

  1. Let u=sin⁡(3x3−5x)u = \sin{\left(3 x^{3} - 5 x \right)}.

  2. Apply the power rule: u8u^{8} goes to 8u78 u^{7}

  3. Then, apply the chain rule. Multiply by ddxsin⁡(3x3−5x)\frac{d}{d x} \sin{\left(3 x^{3} - 5 x \right)}:

    1. Let u=3x3−5xu = 3 x^{3} - 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 ddx(3x3−5x)\frac{d}{d x} \left(3 x^{3} - 5 x\right):

      1. Differentiate 3x3−5x3 x^{3} - 5 x 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: x3x^{3} goes to 3x23 x^{2}

          So, the result is: 9x29 x^{2}

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

          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

          So, the result is: −5-5

        The result is: 9x2−59 x^{2} - 5

      The result of the chain rule is:

      (9x2−5)cos⁡(3x3−5x)\left(9 x^{2} - 5\right) \cos{\left(3 x^{3} - 5 x \right)}

    The result of the chain rule is:

    8⋅(9x2−5)sin⁡7(3x3−5x)cos⁡(3x3−5x)8 \cdot \left(9 x^{2} - 5\right) \sin^{7}{\left(3 x^{3} - 5 x \right)} \cos{\left(3 x^{3} - 5 x \right)}

  4. Now simplify:

    (72x2−40)sin⁡7(x(3x2−5))cos⁡(x(3x2−5))\left(72 x^{2} - 40\right) \sin^{7}{\left(x \left(3 x^{2} - 5\right) \right)} \cos{\left(x \left(3 x^{2} - 5\right) \right)}

The answer is:

(72x2−40)sin⁡7(x(3x2−5))cos⁡(x(3x2−5))\left(72 x^{2} - 40\right) \sin^{7}{\left(x \left(3 x^{2} - 5\right) \right)} \cos{\left(x \left(3 x^{2} - 5\right) \right)}

The graph
02468-8-6-4-2-1010-50005000
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The first derivative [src] 
     7/   3      \ /        2\    /   3      \
8*sin \3*x  - 5*x/*\-5 + 9*x /*cos\3*x  - 5*x/
8⋅(9x2−5)sin⁡7(3x3−5x)cos⁡(3x3−5x)8 \cdot \left(9 x^{2} - 5\right) \sin^{7}{\left(3 x^{3} - 5 x \right)} \cos{\left(3 x^{3} - 5 x \right)}
Simplify
The second derivative [src] 
                      /             2                                    2                                                                 \
     6/  /        2\\ |  /        2\     2/  /        2\\     /        2\     2/  /        2\\           /  /        2\\    /  /        2\\|
8*sin \x*\-5 + 3*x //*\- \-5 + 9*x / *sin \x*\-5 + 3*x // + 7*\-5 + 9*x / *cos \x*\-5 + 3*x // + 18*x*cos\x*\-5 + 3*x //*sin\x*\-5 + 3*x ///
8⋅(−(9x2−5)2sin⁡2(x(3x2−5))+7(9x2−5)2cos⁡2(x(3x2−5))+18xsin⁡(x(3x2−5))cos⁡(x(3x2−5)))sin⁡6(x(3x2−5))8 \cdot \left(- \left(9 x^{2} - 5\right)^{2} \sin^{2}{\left(x \left(3 x^{2} - 5\right) \right)} + 7 \left(9 x^{2} - 5\right)^{2} \cos^{2}{\left(x \left(3 x^{2} - 5\right) \right)} + 18 x \sin{\left(x \left(3 x^{2} - 5\right) \right)} \cos{\left(x \left(3 x^{2} - 5\right) \right)}\right) \sin^{6}{\left(x \left(3 x^{2} - 5\right) \right)}
Simplify
The third derivative [src] 
                       /                                                         3                                                                            3                                                                                                  \
      5/  /        2\\ |     2/  /        2\\    /  /        2\\      /        2\     3/  /        2\\           3/  /        2\\ /        2\      /        2\     2/  /        2\\    /  /        2\\            2/  /        2\\ /        2\    /  /        2\\|
16*sin \x*\-5 + 3*x //*\9*sin \x*\-5 + 3*x //*cos\x*\-5 + 3*x // + 21*\-5 + 9*x / *cos \x*\-5 + 3*x // - 27*x*sin \x*\-5 + 3*x //*\-5 + 9*x / - 11*\-5 + 9*x / *sin \x*\-5 + 3*x //*cos\x*\-5 + 3*x // + 189*x*cos \x*\-5 + 3*x //*\-5 + 9*x /*sin\x*\-5 + 3*x ///
16(−11(9x2−5)3sin⁡2(x(3x2−5))cos⁡(x(3x2−5))+21(9x2−5)3cos⁡3(x(3x2−5))−27x(9x2−5)sin⁡3(x(3x2−5))+189x(9x2−5)sin⁡(x(3x2−5))cos⁡2(x(3x2−5))+9sin⁡2(x(3x2−5))cos⁡(x(3x2−5)))sin⁡5(x(3x2−5))16 \left(- 11 \left(9 x^{2} - 5\right)^{3} \sin^{2}{\left(x \left(3 x^{2} - 5\right) \right)} \cos{\left(x \left(3 x^{2} - 5\right) \right)} + 21 \left(9 x^{2} - 5\right)^{3} \cos^{3}{\left(x \left(3 x^{2} - 5\right) \right)} - 27 x \left(9 x^{2} - 5\right) \sin^{3}{\left(x \left(3 x^{2} - 5\right) \right)} + 189 x \left(9 x^{2} - 5\right) \sin{\left(x \left(3 x^{2} - 5\right) \right)} \cos^{2}{\left(x \left(3 x^{2} - 5\right) \right)} + 9 \sin^{2}{\left(x \left(3 x^{2} - 5\right) \right)} \cos{\left(x \left(3 x^{2} - 5\right) \right)}\right) \sin^{5}{\left(x \left(3 x^{2} - 5\right) \right)}
Simplify
The graph
Derivative of sin^8(3x^3-5x)
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