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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
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
log(x)*sin(2*x + 5)
$$\log{\left(x \right)} \sin{\left(2 x + 5 \right)}$$ 
log(x)*sin(2*x + 5)
Detail solution

  1. Apply the product rule:

    ; to find :

    1. The derivative of is .

    ; to find :

    1. Let .

    2. The derivative of sine is cosine:

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

      1. Differentiate 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: goes to

          So, the result is:

        2. The derivative of the constant is zero.

        The result is:

      The result of the chain rule 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] 
sin(2*x + 5)                        
------------ + 2*cos(2*x + 5)*log(x)
     x
$$2 \log{\left(x \right)} \cos{\left(2 x + 5 \right)} + \frac{\sin{\left(2 x + 5 \right)}}{x}$$
Simplify
The second derivative [src] 
  sin(5 + 2*x)                           4*cos(5 + 2*x)
- ------------ - 4*log(x)*sin(5 + 2*x) + --------------
        2                                      x       
       x
$$- 4 \log{\left(x \right)} \sin{\left(2 x + 5 \right)} + \frac{4 \cos{\left(2 x + 5 \right)}}{x} - \frac{\sin{\left(2 x + 5 \right)}}{x^{2}}$$
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       /
$$2 \left(- 4 \log{\left(x \right)} \cos{\left(2 x + 5 \right)} - \frac{6 \sin{\left(2 x + 5 \right)}}{x} - \frac{3 \cos{\left(2 x + 5 \right)}}{x^{2}} + \frac{\sin{\left(2 x + 5 \right)}}{x^{3}}\right)$$
Simplify
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