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The derivative of the function/ ln*sin*(2*x+5)

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

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

You have entered [src] 
log(x)*sin(2*x + 5)
log⁡(x)sin⁡(2x+5)\log{\left(x \right)} \sin{\left(2 x + 5 \right)} 
log(x)*sin(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)=log⁡(x)f{\left(x \right)} = \log{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of log⁡(x)\log{\left(x \right)} is 1x\frac{1}{x}.

    g(x)=sin⁡(2x+5)g{\left(x \right)} = \sin{\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 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(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:

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

    The result is: 2log⁡(x)cos⁡(2x+5)+sin⁡(2x+5)x2 \log{\left(x \right)} \cos{\left(2 x + 5 \right)} + \frac{\sin{\left(2 x + 5 \right)}}{x}

  2. Now simplify:

    2log⁡(x)cos⁡(2x+5)+sin⁡(2x+5)x2 \log{\left(x \right)} \cos{\left(2 x + 5 \right)} + \frac{\sin{\left(2 x + 5 \right)}}{x}

The answer is:

2log⁡(x)cos⁡(2x+5)+sin⁡(2x+5)x2 \log{\left(x \right)} \cos{\left(2 x + 5 \right)} + \frac{\sin{\left(2 x + 5 \right)}}{x}

The graph
02468-8-6-4-2-1010-2020
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
sin(2*x + 5)                        
------------ + 2*cos(2*x + 5)*log(x)
     x
2log⁡(x)cos⁡(2x+5)+sin⁡(2x+5)x2 \log{\left(x \right)} \cos{\left(2 x + 5 \right)} + \frac{\sin{\left(2 x + 5 \right)}}{x}
Simplify
The second derivative [src] 
  sin(5 + 2*x)                           4*cos(5 + 2*x)
- ------------ - 4*log(x)*sin(5 + 2*x) + --------------
        2                                      x       
       x
−4log⁡(x)sin⁡(2x+5)+4cos⁡(2x+5)x−sin⁡(2x+5)x2- 4 \log{\left(x \right)} \sin{\left(2 x + 5 \right)} + \frac{4 \cos{\left(2 x + 5 \right)}}{x} - \frac{\sin{\left(2 x + 5 \right)}}{x^{2}}
Simplify
The third derivative [src] 
  /sin(5 + 2*x)   6*sin(5 + 2*x)                           3*cos(5 + 2*x)\
2*|------------ - -------------- - 4*cos(5 + 2*x)*log(x) - --------------|
  |      3              x                                         2      |
  \     x                                                        x       /
2(−4log⁡(x)cos⁡(2x+5)−6sin⁡(2x+5)x−3cos⁡(2x+5)x2+sin⁡(2x+5)x3)2 \left(- 4 \log{\left(x \right)} \cos{\left(2 x + 5 \right)} - \frac{6 \sin{\left(2 x + 5 \right)}}{x} - \frac{3 \cos{\left(2 x + 5 \right)}}{x^{2}} + \frac{\sin{\left(2 x + 5 \right)}}{x^{3}}\right)
Simplify
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