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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
v

The graph:

from to

Piecewise:

The solution

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

  1. Differentiate 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 is .

      So, the result is:

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

      1. Let .

      2. The derivative of is .

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of sine is cosine:

        The result of the chain rule is:

      So, the result is:

    The result is:

  2. Now simplify:

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
1   2*cos(x)
- + --------
x    sin(x)
$$\frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{1}{x}$$
Simplify
The second derivative [src] 
 /              2   \
 |    1    2*cos (x)|
-|2 + -- + ---------|
 |     2       2    |
 \    x     sin (x) /
$$- (2 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{1}{x^{2}})$$
Simplify
The third derivative [src] 
  /          3              \
  |1    2*cos (x)   2*cos(x)|
2*|-- + --------- + --------|
  | 3       3        sin(x) |
  \x     sin (x)            /
$$2 \left(\frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} + \frac{1}{x^{3}}\right)$$
Simplify
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