Mister Exam

Other calculators

Derivative of sqrt(sin5x)*e^cos3x

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  __________  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
  1. Apply the product rule:

    ; to find :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. Let .

      2. The derivative of sine is cosine:

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result of the chain rule is:

      The result of the chain rule is:

    ; to find :

    1. Let .

    2. The derivative of is itself.

    3. Then, apply the chain rule. Multiply by :

      1. Let .

      2. The derivative of cosine is negative sine:

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result of the chain rule is:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The graph
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)    
$$- 3 e^{\cos{\left(3 x \right)}} \sin{\left(3 x \right)} \sqrt{\sin{\left(5 x \right)}} + \frac{5 e^{\cos{\left(3 x \right)}} \cos{\left(5 x \right)}}{2 \sqrt{\sin{\left(5 x \right)}}}$$
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)     /          
$$\left(9 \left(\sin^{2}{\left(3 x \right)} - \cos{\left(3 x \right)}\right) \sqrt{\sin{\left(5 x \right)}} - \frac{15 \sin{\left(3 x \right)} \cos{\left(5 x \right)}}{\sqrt{\sin{\left(5 x \right)}}} - \frac{25 \sqrt{\sin{\left(5 x \right)}}}{2} - \frac{25 \cos^{2}{\left(5 x \right)}}{4 \sin^{\frac{3}{2}}{\left(5 x \right)}}\right) e^{\cos{\left(3 x \right)}}$$
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)           /          
$$\left(\frac{125 \left(2 + \frac{3 \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}\right) \cos{\left(5 x \right)}}{8 \sqrt{\sin{\left(5 x \right)}}} + \frac{135 \left(\sin^{2}{\left(3 x \right)} - \cos{\left(3 x \right)}\right) \cos{\left(5 x \right)}}{2 \sqrt{\sin{\left(5 x \right)}}} + \frac{225 \left(2 \sqrt{\sin{\left(5 x \right)}} + \frac{\cos^{2}{\left(5 x \right)}}{\sin^{\frac{3}{2}}{\left(5 x \right)}}\right) \sin{\left(3 x \right)}}{4} + 27 \left(- \sin^{2}{\left(3 x \right)} + 3 \cos{\left(3 x \right)} + 1\right) \sin{\left(3 x \right)} \sqrt{\sin{\left(5 x \right)}}\right) e^{\cos{\left(3 x \right)}}$$