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Identical expressions
(e^sin(x)+ three *x)^ three
(e to the power of sinus of (x) plus 3 multiply by x) cubed
(e to the power of sinus of (x) plus three multiply by x) to the power of three
(esin(x)+3*x)3
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(e to the power of sin(x)+3*x) to the power of 3
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e^sinx+3x^3
Similar expressions
(e^sin(x)-3*x)^3
(e^sinx+3*x)^3

The derivative of the function/ (e^sin(x)+3*x)^3

Derivative of (e^sin(x)+3*x)^3

The solution

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               3
/ sin(x)      \ 
\E       + 3*x/

$$\left(e^{\sin{\left(x \right)}} + 3 x\right)^{3}$$

(E^sin(x) + 3*x)^3

Detail solution

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

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

$3 \left(3 x + e^{\sin{\left(x \right)}}\right)^{2} \left(e^{\sin{\left(x \right)}} \cos{\left(x \right)} + 3\right)$

The answer is:

$3 \left(3 x + e^{\sin{\left(x \right)}}\right)^{2} \left(e^{\sin{\left(x \right)}} \cos{\left(x \right)} + 3\right)$

The graph

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The first derivative [src]

               2                       
/ sin(x)      \  /              sin(x)\
\E       + 3*x/ *\9 + 3*cos(x)*e      /

$$\left(e^{\sin{\left(x \right)}} + 3 x\right)^{2} \left(3 e^{\sin{\left(x \right)}} \cos{\left(x \right)} + 9\right)$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                      3                  2                                                                                                              \
  |  /            sin(x)\    /       sin(x)\  /       2              \         sin(x)     /            sin(x)\ /     2            \ /       sin(x)\  sin(x)|
3*\2*\3 + cos(x)*e      /  - \3*x + e      / *\1 - cos (x) + 3*sin(x)/*cos(x)*e       - 6*\3 + cos(x)*e      /*\- cos (x) + sin(x)/*\3*x + e      /*e      /

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

Simplify

The graph

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Derivative of (e^sin(x)+3*x)^3

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The solution