Derivative of sin(x)/(x+1)

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sin(x)
------
x + 1

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

sin(x)/(x + 1)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = x + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

cos(x)    sin(x) 
------ - --------
x + 1           2
         (x + 1)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

          6*sin(x)   3*sin(x)   6*cos(x)
-cos(x) - -------- + -------- + --------
                 3    1 + x            2
          (1 + x)               (1 + x) 
----------------------------------------
                 1 + x

$$\frac{- \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{x + 1} + \frac{6 \cos{\left(x \right)}}{\left(x + 1\right)^{2}} - \frac{6 \sin{\left(x \right)}}{\left(x + 1\right)^{3}}}{x + 1}$$

Simplify

The graph

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Derivative of sin(x)/(x+1)

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The solution