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Derivative of sin(x-1)+ln((2x+3)/5)

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                /2*x + 3\
sin(x - 1) + log|-------|
                \   5   /

$$\log{\left(\frac{2 x + 3}{5} \right)} + \sin{\left(x - 1 \right)}$$

sin(x - 1) + log((2*x + 3)/5)

Detail solution

Differentiate $\log{\left(\frac{2 x + 3}{5} \right)} + \sin{\left(x - 1 \right)}$ term by term:
1. Let $u = x - 1$ .
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 $\frac{d}{d x} \left(x - 1\right)$ :
  1. Differentiate $x - 1$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of the constant $-1$ is zero.
    The result is: $1$
  The result of the chain rule is:
  
  $\cos{\left(x - 1 \right)}$
4. Let $u = \frac{2 x + 3}{5}$ .
5. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
6. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{2 x + 3}{5}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Differentiate $2 x + 3$ term by term:
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
      2. The derivative of the constant $3$ is zero.
      The result is: $2$
    So, the result is: $\frac{2}{5}$
  The result of the chain rule is:
  
  $\frac{2}{2 x + 3}$
The result is: $\cos{\left(x - 1 \right)} + \frac{2}{2 x + 3}$
Now simplify:

$\frac{\left(2 x + 3\right) \cos{\left(x - 1 \right)} + 2}{2 x + 3}$

The answer is:

$\frac{\left(2 x + 3\right) \cos{\left(x - 1 \right)} + 2}{2 x + 3}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2                
------- + cos(x - 1)
2*x + 3

$$\cos{\left(x - 1 \right)} + \frac{2}{2 x + 3}$$

Simplify

The second derivative [src]

 /    4                   \
-|---------- + sin(-1 + x)|
 |         2              |
 \(3 + 2*x)               /

$$- (\sin{\left(x - 1 \right)} + \frac{4}{\left(2 x + 3\right)^{2}})$$

Simplify

The third derivative [src]

                   16    
-cos(-1 + x) + ----------
                        3
               (3 + 2*x)

$$- \cos{\left(x - 1 \right)} + \frac{16}{\left(2 x + 3\right)^{3}}$$

Simplify