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Identical expressions
ln*sin*(two *x+ five)
ln multiply by sinus of multiply by (2 multiply by x plus 5)
ln multiply by sinus of multiply by (two multiply by x plus five)
lnsin(2x+5)
lnsin2x+5
Similar expressions
ln(sin(2x+5))
ln*sin*(2*x-5)
y=lnsin2x+5
y=lnsin(2x+5)

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

Derivative of lnsin(2*x+5)

The solution

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log(x)*sin(2*x + 5)

$$\log{\left(x \right)} \sin{\left(2 x + 5 \right)}$$

log(x)*sin(2*x + 5)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
$g{\left(x \right)} = \sin{\left(2 x + 5 \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x + 5$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 5\right)$ :
  1. Differentiate $2 x + 5$ 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: $x$ goes to $1$
      So, the result is: $2$
    2. The derivative of the constant $5$ is zero.
    The result is: $2$
  The result of the chain rule is:
  
  $2 \cos{\left(2 x + 5 \right)}$
The result is: $2 \log{\left(x \right)} \cos{\left(2 x + 5 \right)} + \frac{\sin{\left(2 x + 5 \right)}}{x}$
Now simplify:

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

The answer is:

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

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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Derivative of lnsin(2*x+5)

The graph:

Piecewise:

The solution

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

The graph:

Piecewise:

The solution

Derivative of lnsin(2*x+5)