Derivative of sin((x+3)/2)

You have entered [src]

   /x + 3\
sin|-----|
   \  2  /

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

sin((x + 3)/2)

Detail solution

Let $u = \frac{x + 3}{2}$ .
The derivative of sine is cosine:

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

$\frac{\cos{\left(\frac{x + 3}{2} \right)}}{2}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   /x + 3\
cos|-----|
   \  2  /
----------
    2

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

Simplify

The second derivative [src]

    /3 + x\ 
-sin|-----| 
    \  2  / 
------------
     4

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

Simplify

The third derivative [src]

    /3 + x\ 
-cos|-----| 
    \  2  / 
------------
     8

$$- \frac{\cos{\left(\frac{x + 3}{2} \right)}}{8}$$

Simplify