Mister Exam

Derivative of y=e^(sinx)cos3x

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
 sin(x)         
E      *cos(3*x)
$$e^{\sin{\left(x \right)}} \cos{\left(3 x \right)}$$
E^sin(x)*cos(3*x)
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Let .

    2. The derivative of is itself.

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

      1. The derivative of sine is cosine:

      The result of the chain rule is:

    ; to find :

    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 is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
     sin(x)                             sin(x)
- 3*e      *sin(3*x) + cos(x)*cos(3*x)*e      
$$- 3 e^{\sin{\left(x \right)}} \sin{\left(3 x \right)} + e^{\sin{\left(x \right)}} \cos{\left(x \right)} \cos{\left(3 x \right)}$$
The second derivative [src]
 /             /     2            \                             \  sin(x)
-\9*cos(3*x) + \- cos (x) + sin(x)/*cos(3*x) + 6*cos(x)*sin(3*x)/*e      
$$- \left(\left(\sin{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos{\left(3 x \right)} + 6 \sin{\left(3 x \right)} \cos{\left(x \right)} + 9 \cos{\left(3 x \right)}\right) e^{\sin{\left(x \right)}}$$
The third derivative [src]
/                                     /     2            \            /       2              \                \  sin(x)
\27*sin(3*x) - 27*cos(x)*cos(3*x) + 9*\- cos (x) + sin(x)/*sin(3*x) - \1 - cos (x) + 3*sin(x)/*cos(x)*cos(3*x)/*e      
$$\left(9 \left(\sin{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin{\left(3 x \right)} - \left(3 \sin{\left(x \right)} - \cos^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \cos{\left(3 x \right)} + 27 \sin{\left(3 x \right)} - 27 \cos{\left(x \right)} \cos{\left(3 x \right)}\right) e^{\sin{\left(x \right)}}$$
The graph
Derivative of y=e^(sinx)cos3x