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Derivative of 5*x^4-7*x^2-x
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Derivative of 4*sqrt(3)*x/3
Identical expressions
sqrt(sin5x)*e^cos3x
square root of ( sinus of 5x) multiply by e to the power of co sinus of e of 3x
√(sin5x)*e^cos3x
sqrt(sin5x)*ecos3x
sqrtsin5x*ecos3x
sqrt(sin5x)e^cos3x
sqrt(sin5x)ecos3x
sqrtsin5xecos3x
sqrtsin5xe^cos3x

The derivative of the function/ sqrt(sin5x)*e^cos3x

Derivative of sqrt(sin5x)*e^cos3x

The solution

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  __________  cos(3*x)
\/ sin(5*x) *E

$$e^{\cos{\left(3 x \right)}} \sqrt{\sin{\left(5 x \right)}}$$

sqrt(sin(5*x))*E^cos(3*x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sqrt{\sin{\left(5 x \right)}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(5 x \right)}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(5 x \right)}$ :
  1. Let $u = 5 x$ .
  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 $\frac{d}{d x} 5 x$ :
    1. 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: $5$
    The result of the chain rule is:
    
    $5 \cos{\left(5 x \right)}$
  The result of the chain rule is:
  
  $\frac{5 \cos{\left(5 x \right)}}{2 \sqrt{\sin{\left(5 x \right)}}}$
$g{\left(x \right)} = e^{\cos{\left(3 x \right)}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \cos{\left(3 x \right)}$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(3 x \right)}$ :
  1. Let $u = 3 x$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
    1. 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 of the chain rule is:
    
    $- 3 \sin{\left(3 x \right)}$
  The result of the chain rule is:
  
  $- 3 e^{\cos{\left(3 x \right)}} \sin{\left(3 x \right)}$
The result is: $- 3 e^{\cos{\left(3 x \right)}} \sin{\left(3 x \right)} \sqrt{\sin{\left(5 x \right)}} + \frac{5 e^{\cos{\left(3 x \right)}} \cos{\left(5 x \right)}}{2 \sqrt{\sin{\left(5 x \right)}}}$
Now simplify:

$\frac{\left(- 6 \sin{\left(3 x \right)} \sin{\left(5 x \right)} + 5 \cos{\left(5 x \right)}\right) e^{\cos{\left(3 x \right)}}}{2 \sqrt{\sin{\left(5 x \right)}}}$

The answer is:

$\frac{\left(- 6 \sin{\left(3 x \right)} \sin{\left(5 x \right)} + 5 \cos{\left(5 x \right)}\right) e^{\cos{\left(3 x \right)}}}{2 \sqrt{\sin{\left(5 x \right)}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                                                  cos(3*x)
      __________  cos(3*x)            5*cos(5*x)*e        
- 3*\/ sin(5*x) *e        *sin(3*x) + --------------------
                                             __________   
                                         2*\/ sin(5*x)

$$- 3 e^{\cos{\left(3 x \right)}} \sin{\left(3 x \right)} \sqrt{\sin{\left(5 x \right)}} + \frac{5 e^{\cos{\left(3 x \right)}} \cos{\left(5 x \right)}}{2 \sqrt{\sin{\left(5 x \right)}}}$$

Simplify

The second derivative [src]

/       __________                                                  2                            \          
|  25*\/ sin(5*x)        __________ /   2                \    25*cos (5*x)   15*cos(5*x)*sin(3*x)|  cos(3*x)
|- --------------- + 9*\/ sin(5*x) *\sin (3*x) - cos(3*x)/ - ------------- - --------------------|*e        
|         2                                                       3/2              __________    |          
\                                                            4*sin   (5*x)       \/ sin(5*x)     /

$$\left(9 \left(\sin^{2}{\left(3 x \right)} - \cos{\left(3 x \right)}\right) \sqrt{\sin{\left(5 x \right)}} - \frac{15 \sin{\left(3 x \right)} \cos{\left(5 x \right)}}{\sqrt{\sin{\left(5 x \right)}}} - \frac{25 \sqrt{\sin{\left(5 x \right)}}}{2} - \frac{25 \cos^{2}{\left(5 x \right)}}{4 \sin^{\frac{3}{2}}{\left(5 x \right)}}\right) e^{\cos{\left(3 x \right)}}$$

Simplify

The third derivative [src]

/    /                     2      \                                                                        /         2     \                                               \          
|    |    __________    cos (5*x) |                                                                        |    3*cos (5*x)|                                               |          
|225*|2*\/ sin(5*x)  + -----------|*sin(3*x)                                                           125*|2 + -----------|*cos(5*x)                                      |          
|    |                    3/2     |                                                                        |        2      |                /   2                \         |          
|    \                 sin   (5*x)/                 __________ /       2                  \                \     sin (5*x) /            135*\sin (3*x) - cos(3*x)/*cos(5*x)|  cos(3*x)
|------------------------------------------- + 27*\/ sin(5*x) *\1 - sin (3*x) + 3*cos(3*x)/*sin(3*x) + ------------------------------ + -----------------------------------|*e        
|                     4                                                                                            __________                          __________          |          
\                                                                                                              8*\/ sin(5*x)                       2*\/ sin(5*x)           /

$$\left(\frac{125 \left(2 + \frac{3 \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}\right) \cos{\left(5 x \right)}}{8 \sqrt{\sin{\left(5 x \right)}}} + \frac{135 \left(\sin^{2}{\left(3 x \right)} - \cos{\left(3 x \right)}\right) \cos{\left(5 x \right)}}{2 \sqrt{\sin{\left(5 x \right)}}} + \frac{225 \left(2 \sqrt{\sin{\left(5 x \right)}} + \frac{\cos^{2}{\left(5 x \right)}}{\sin^{\frac{3}{2}}{\left(5 x \right)}}\right) \sin{\left(3 x \right)}}{4} + 27 \left(- \sin^{2}{\left(3 x \right)} + 3 \cos{\left(3 x \right)} + 1\right) \sin{\left(3 x \right)} \sqrt{\sin{\left(5 x \right)}}\right) e^{\cos{\left(3 x \right)}}$$

Simplify

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Derivative of sqrt(sin5x)*e^cos3x

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The solution