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Similar expressions
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The derivative of the function/ 3/x-3-2sin(x-1)

Derivative of 3/x-3-2sin(x-1)

The solution

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3                   
- - 3 - 2*sin(x - 1)
x

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

3/x - 3 - 2*sin(x - 1)

Detail solution

Differentiate $\left(-3 + \frac{3}{x}\right) - 2 \sin{\left(x - 1 \right)}$ term by term:
1. Differentiate $-3 + \frac{3}{x}$ 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: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
    So, the result is: $- \frac{3}{x^{2}}$
  2. The derivative of the constant $-3$ is zero.
  The result is: $- \frac{3}{x^{2}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  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:
    
    $\cos{\left(x - 1 \right)}$
  So, the result is: $- 2 \cos{\left(x - 1 \right)}$
The result is: $- 2 \cos{\left(x - 1 \right)} - \frac{3}{x^{2}}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /  9               \
2*|- -- + cos(-1 + x)|
  |   4              |
  \  x               /

$$2 \left(\cos{\left(x - 1 \right)} - \frac{9}{x^{4}}\right)$$

Simplify