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Derivative of (sin8x*ln(2x+5))

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

You have entered [src]
sin(8*x)*log(2*x + 5)
log(2x+5)sin(8x)\log{\left(2 x + 5 \right)} \sin{\left(8 x \right)}
sin(8*x)*log(2*x + 5)
Detail solution
  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=sin(8x)f{\left(x \right)} = \sin{\left(8 x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=8xu = 8 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 ddx8x\frac{d}{d x} 8 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: 88

      The result of the chain rule is:

      8cos(8x)8 \cos{\left(8 x \right)}

    g(x)=log(2x+5)g{\left(x \right)} = \log{\left(2 x + 5 \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=2x+5u = 2 x + 5.

    2. The derivative of log(u)\log{\left(u \right)} is 1u\frac{1}{u}.

    3. Then, apply the chain rule. Multiply by ddx(2x+5)\frac{d}{d x} \left(2 x + 5\right):

      1. Differentiate 2x+52 x + 5 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: xx goes to 11

          So, the result is: 22

        2. The derivative of the constant 55 is zero.

        The result is: 22

      The result of the chain rule is:

      22x+5\frac{2}{2 x + 5}

    The result is: 8log(2x+5)cos(8x)+2sin(8x)2x+58 \log{\left(2 x + 5 \right)} \cos{\left(8 x \right)} + \frac{2 \sin{\left(8 x \right)}}{2 x + 5}

  2. Now simplify:

    2(4(2x+5)log(2x+5)cos(8x)+sin(8x))2x+5\frac{2 \left(4 \left(2 x + 5\right) \log{\left(2 x + 5 \right)} \cos{\left(8 x \right)} + \sin{\left(8 x \right)}\right)}{2 x + 5}


The answer is:

2(4(2x+5)log(2x+5)cos(8x)+sin(8x))2x+5\frac{2 \left(4 \left(2 x + 5\right) \log{\left(2 x + 5 \right)} \cos{\left(8 x \right)} + \sin{\left(8 x \right)}\right)}{2 x + 5}

The graph
02468-8-6-4-2-1010-5050
The first derivative [src]
2*sin(8*x)                          
---------- + 8*cos(8*x)*log(2*x + 5)
 2*x + 5                            
8log(2x+5)cos(8x)+2sin(8x)2x+58 \log{\left(2 x + 5 \right)} \cos{\left(8 x \right)} + \frac{2 \sin{\left(8 x \right)}}{2 x + 5}
The second derivative [src]
  /   sin(8*x)                               8*cos(8*x)\
4*|- ---------- - 16*log(5 + 2*x)*sin(8*x) + ----------|
  |           2                               5 + 2*x  |
  \  (5 + 2*x)                                         /
4(16log(2x+5)sin(8x)+8cos(8x)2x+5sin(8x)(2x+5)2)4 \left(- 16 \log{\left(2 x + 5 \right)} \sin{\left(8 x \right)} + \frac{8 \cos{\left(8 x \right)}}{2 x + 5} - \frac{\sin{\left(8 x \right)}}{\left(2 x + 5\right)^{2}}\right)
The third derivative [src]
   / sin(8*x)                               24*sin(8*x)   6*cos(8*x)\
16*|---------- - 32*cos(8*x)*log(5 + 2*x) - ----------- - ----------|
   |         3                                5 + 2*x              2|
   \(5 + 2*x)                                             (5 + 2*x) /
16(32log(2x+5)cos(8x)24sin(8x)2x+56cos(8x)(2x+5)2+sin(8x)(2x+5)3)16 \left(- 32 \log{\left(2 x + 5 \right)} \cos{\left(8 x \right)} - \frac{24 \sin{\left(8 x \right)}}{2 x + 5} - \frac{6 \cos{\left(8 x \right)}}{\left(2 x + 5\right)^{2}} + \frac{\sin{\left(8 x \right)}}{\left(2 x + 5\right)^{3}}\right)