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The derivative of the function/ x*log(2*x)+2^sin(x)

Derivative of x*log(2*x)+2^sin(x)

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
              sin(x)
x*log(2*x) + 2
2sin⁡(x)+xlog⁡(2x)2^{\sin{\left(x \right)}} + x \log{\left(2 x \right)} 
x*log(2*x) + 2^sin(x)
Detail solution

  1. Differentiate 2sin⁡(x)+xlog⁡(2x)2^{\sin{\left(x \right)}} + x \log{\left(2 x \right)} term by term:

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

      1. Apply the power rule: xx goes to 11

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

      1. Let u=2xu = 2 x.

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

      3. Then, apply the chain rule. Multiply by ddx2x\frac{d}{d x} 2 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: 22

        The result of the chain rule is:

        1x\frac{1}{x}

      The result is: log⁡(2x)+1\log{\left(2 x \right)} + 1

    2. Let u=sin⁡(x)u = \sin{\left(x \right)}.

    3. ddu2u=2ulog⁡(2)\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}

    4. Then, apply the chain rule. Multiply by ddxsin⁡(x)\frac{d}{d x} \sin{\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)}

      The result of the chain rule is:

      2sin⁡(x)log⁡(2)cos⁡(x)2^{\sin{\left(x \right)}} \log{\left(2 \right)} \cos{\left(x \right)}

    The result is: 2sin⁡(x)log⁡(2)cos⁡(x)+log⁡(2x)+12^{\sin{\left(x \right)}} \log{\left(2 \right)} \cos{\left(x \right)} + \log{\left(2 x \right)} + 1

  2. Now simplify:

    log⁡(22sin⁡(x)cos⁡(x))+log⁡(2x)+1\log{\left(2^{2^{\sin{\left(x \right)}} \cos{\left(x \right)}} \right)} + \log{\left(2 x \right)} + 1

The answer is:

log⁡(22sin⁡(x)cos⁡(x))+log⁡(2x)+1\log{\left(2^{2^{\sin{\left(x \right)}} \cos{\left(x \right)}} \right)} + \log{\left(2 x \right)} + 1

The graph
02468-8-6-4-2-1010050
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
     sin(x)                         
1 + 2      *cos(x)*log(2) + log(2*x)
2sin⁡(x)log⁡(2)cos⁡(x)+log⁡(2x)+12^{\sin{\left(x \right)}} \log{\left(2 \right)} \cos{\left(x \right)} + \log{\left(2 x \right)} + 1
Simplify
The second derivative [src] 
1    sin(x)    2       2       sin(x)              
- + 2      *cos (x)*log (2) - 2      *log(2)*sin(x)
x
−2sin⁡(x)log⁡(2)sin⁡(x)+2sin⁡(x)log⁡(2)2cos⁡2(x)+1x- 2^{\sin{\left(x \right)}} \log{\left(2 \right)} \sin{\left(x \right)} + 2^{\sin{\left(x \right)}} \log{\left(2 \right)}^{2} \cos^{2}{\left(x \right)} + \frac{1}{x}
Simplify
The third derivative [src] 
  1     sin(x)    3       3       sin(x)                    sin(x)    2                 
- -- + 2      *cos (x)*log (2) - 2      *cos(x)*log(2) - 3*2      *log (2)*cos(x)*sin(x)
   2                                                                                    
  x
−3⋅2sin⁡(x)log⁡(2)2sin⁡(x)cos⁡(x)+2sin⁡(x)log⁡(2)3cos⁡3(x)−2sin⁡(x)log⁡(2)cos⁡(x)−1x2- 3 \cdot 2^{\sin{\left(x \right)}} \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 2^{\sin{\left(x \right)}} \log{\left(2 \right)}^{3} \cos^{3}{\left(x \right)} - 2^{\sin{\left(x \right)}} \log{\left(2 \right)} \cos{\left(x \right)} - \frac{1}{x^{2}}
Simplify
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