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The derivative of the function/ logx\1+2*logsin(x)

Derivative of logx\1+2*logsin(x)

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

You have entered [src] 
log(x)                
------ + 2*log(sin(x))
  1
log⁡(x)1+2log⁡(sin⁡(x))\frac{\log{\left(x \right)}}{1} + 2 \log{\left(\sin{\left(x \right)} \right)} 
log(x)/1 + 2*log(sin(x))
Detail solution

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

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

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

      So, the result is: 1x\frac{1}{x}

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

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

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

      3. 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:

        cos⁡(x)sin⁡(x)\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}

      So, the result is: 2cos⁡(x)sin⁡(x)\frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}

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

  2. Now simplify:

    2tan⁡(x)+1x\frac{2}{\tan{\left(x \right)}} + \frac{1}{x}

The answer is:

2tan⁡(x)+1x\frac{2}{\tan{\left(x \right)}} + \frac{1}{x}

The graph
02468-8-6-4-2-1010-500500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
1   2*cos(x)
- + --------
x    sin(x)
2cos⁡(x)sin⁡(x)+1x\frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{1}{x}
Simplify
The second derivative [src] 
 /              2   \
 |    1    2*cos (x)|
-|2 + -- + ---------|
 |     2       2    |
 \    x     sin (x) /
−(2+2cos⁡2(x)sin⁡2(x)+1x2)- (2 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{1}{x^{2}})
Simplify
The third derivative [src] 
  /          3              \
  |1    2*cos (x)   2*cos(x)|
2*|-- + --------- + --------|
  | 3       3        sin(x) |
  \x     sin (x)            /
2(2cos⁡(x)sin⁡(x)+2cos⁡3(x)sin⁡3(x)+1x3)2 \left(\frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} + \frac{1}{x^{3}}\right)
Simplify
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