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Identical expressions
e^(y/ two +sin(y)/ two + three)
e to the power of (y divide by 2 plus sinus of (y) divide by 2 plus 3)
e to the power of (y divide by two plus sinus of (y) divide by two plus three)
e(y/2+sin(y)/2+3)
ey/2+siny/2+3
e^y/2+siny/2+3
e^(y divide by 2+sin(y) divide by 2+3)
Similar expressions
e^(y/2+sin(y)/2-3)
e^(y/2-sin(y)/2+3)

The derivative of the function/ e^(y/2+sin(y)/2+3)

Derivative of e^(y/2+sin(y)/2+3)

The solution

You have entered [src]

 y   sin(y)    
 - + ------ + 3
 2     2       
e

$$e^{\frac{y}{2} + \frac{\sin{\left(y \right)}}{2} + 3}$$

  / y   sin(y)    \
  | - + ------ + 3|
d | 2     2       |
--\e              /
dy

$$\frac{d}{d y} e^{\frac{y}{2} + \frac{\sin{\left(y \right)}}{2} + 3}$$

Transform

Detail solution

Let $u = \frac{y}{2} + \frac{\sin{\left(y \right)}}{2} + 3$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d y} \left(\frac{y}{2} + \frac{\sin{\left(y \right)}}{2} + 3\right)$ :
1. Differentiate $\frac{y}{2} + \frac{\sin{\left(y \right)}}{2} + 3$ 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: $y$ goes to $1$
    So, the result is: $\frac{1}{2}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of sine is cosine:
      
      $\frac{d}{d y} \sin{\left(y \right)} = \cos{\left(y \right)}$
    So, the result is: $\frac{\cos{\left(y \right)}}{2}$
  3. The derivative of the constant $3$ is zero.
  The result is: $\frac{\cos{\left(y \right)}}{2} + \frac{1}{2}$
The result of the chain rule is:

$\left(\frac{\cos{\left(y \right)}}{2} + \frac{1}{2}\right) e^{\frac{y}{2} + \frac{\sin{\left(y \right)}}{2} + 3}$
Now simplify:

$\frac{\left(\cos{\left(y \right)} + 1\right) e^{\frac{y}{2} + \frac{\sin{\left(y \right)}}{2} + 3}}{2}$

The answer is:

$\frac{\left(\cos{\left(y \right)} + 1\right) e^{\frac{y}{2} + \frac{\sin{\left(y \right)}}{2} + 3}}{2}$

The graph

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The first derivative [src]

              y   sin(y)    
              - + ------ + 3
/1   cos(y)\  2     2       
|- + ------|*e              
\2     2   /

$$\left(\frac{\cos{\left(y \right)}}{2} + \frac{1}{2}\right) e^{\frac{y}{2} + \frac{\sin{\left(y \right)}}{2} + 3}$$

Simplify

The second derivative [src]

                                y   sin(y)
                            3 + - + ------
/            2           \      2     2   
\(1 + cos(y))  - 2*sin(y)/*e              
------------------------------------------
                    4

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

Simplify

The third derivative [src]

                                                        y   sin(y)
                                                    3 + - + ------
/            3                                   \      2     2   
\(1 + cos(y))  - 4*cos(y) - 6*(1 + cos(y))*sin(y)/*e              
------------------------------------------------------------------
                                8

$$\frac{\left(\left(\cos{\left(y \right)} + 1\right)^{3} - 6 \left(\cos{\left(y \right)} + 1\right) \sin{\left(y \right)} - 4 \cos{\left(y \right)}\right) e^{\frac{y}{2} + \frac{\sin{\left(y \right)}}{2} + 3}}{8}$$

Simplify

The graph

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Derivative of e^(y/2+sin(y)/2+3)

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The solution