Mister Exam

Other calculators


cos(5x)*e^(x/2)

Derivative of cos(5x)*e^(x/2)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
          x
          -
          2
cos(5*x)*e 
ex2cos(5x)e^{\frac{x}{2}} \cos{\left(5 x \right)}
  /          x\
  |          -|
d |          2|
--\cos(5*x)*e /
dx             
ddxex2cos(5x)\frac{d}{d x} e^{\frac{x}{2}} \cos{\left(5 x \right)}
Detail solution
  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=cos(5x)f{\left(x \right)} = \cos{\left(5 x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=5xu = 5 x.

    2. The derivative of cosine is negative sine:

      dducos(u)=sin(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

    3. Then, apply the chain rule. Multiply by ddx5x\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: xx goes to 11

        So, the result is: 55

      The result of the chain rule is:

      5sin(5x)- 5 \sin{\left(5 x \right)}

    g(x)=ex2g{\left(x \right)} = e^{\frac{x}{2}}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=x2u = \frac{x}{2}.

    2. The derivative of eue^{u} is itself.

    3. Then, apply the chain rule. Multiply by ddxx2\frac{d}{d x} \frac{x}{2}:

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

        1. Apply the power rule: xx goes to 11

        So, the result is: 12\frac{1}{2}

      The result of the chain rule is:

      ex22\frac{e^{\frac{x}{2}}}{2}

    The result is: 5ex2sin(5x)+ex2cos(5x)2- 5 e^{\frac{x}{2}} \sin{\left(5 x \right)} + \frac{e^{\frac{x}{2}} \cos{\left(5 x \right)}}{2}

  2. Now simplify:

    (10sin(5x)+cos(5x))ex22\frac{\left(- 10 \sin{\left(5 x \right)} + \cos{\left(5 x \right)}\right) e^{\frac{x}{2}}}{2}


The answer is:

(10sin(5x)+cos(5x))ex22\frac{\left(- 10 \sin{\left(5 x \right)} + \cos{\left(5 x \right)}\right) e^{\frac{x}{2}}}{2}

The graph
02468-8-6-4-2-1010-10001000
The first derivative [src]
          x                
          -      x         
          2      -         
cos(5*x)*e       2         
----------- - 5*e *sin(5*x)
     2                     
5ex2sin(5x)+ex2cos(5x)2- 5 e^{\frac{x}{2}} \sin{\left(5 x \right)} + \frac{e^{\frac{x}{2}} \cos{\left(5 x \right)}}{2}
The second derivative [src]
                             x
                             -
/              99*cos(5*x)\  2
|-5*sin(5*x) - -----------|*e 
\                   4     /   
(5sin(5x)99cos(5x)4)ex2\left(- 5 \sin{\left(5 x \right)} - \frac{99 \cos{\left(5 x \right)}}{4}\right) e^{\frac{x}{2}}
The third derivative [src]
                                x
                                -
                                2
(-299*cos(5*x) + 970*sin(5*x))*e 
---------------------------------
                8                
(970sin(5x)299cos(5x))ex28\frac{\left(970 \sin{\left(5 x \right)} - 299 \cos{\left(5 x \right)}\right) e^{\frac{x}{2}}}{8}
The graph
Derivative of cos(5x)*e^(x/2)