Mister Exam

Derivative of е^sinx*sinx

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
 sin(x)       
E      *sin(x)
esin(x)sin(x)e^{\sin{\left(x \right)}} \sin{\left(x \right)}
E^sin(x)*sin(x)
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)=esin(x)f{\left(x \right)} = e^{\sin{\left(x \right)}}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=sin(x)u = \sin{\left(x \right)}.

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

    3. Then, apply the chain rule. Multiply by ddxsin(x)\frac{d}{d x} \sin{\left(x \right)}:

      1. The derivative of sine is cosine:

        ddxsin(x)=cos(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

      The result of the chain rule is:

      esin(x)cos(x)e^{\sin{\left(x \right)}} \cos{\left(x \right)}

    g(x)=sin(x)g{\left(x \right)} = \sin{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. The derivative of sine is cosine:

      ddxsin(x)=cos(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

    The result is: esin(x)sin(x)cos(x)+esin(x)cos(x)e^{\sin{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)} + e^{\sin{\left(x \right)}} \cos{\left(x \right)}

  2. Now simplify:

    (sin(x)+1)esin(x)cos(x)\left(\sin{\left(x \right)} + 1\right) e^{\sin{\left(x \right)}} \cos{\left(x \right)}


The answer is:

(sin(x)+1)esin(x)cos(x)\left(\sin{\left(x \right)} + 1\right) e^{\sin{\left(x \right)}} \cos{\left(x \right)}

The graph
02468-8-6-4-2-10105-5
The first derivative [src]
        sin(x)           sin(x)       
cos(x)*e       + cos(x)*e      *sin(x)
esin(x)sin(x)cos(x)+esin(x)cos(x)e^{\sin{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)} + e^{\sin{\left(x \right)}} \cos{\left(x \right)}
The second derivative [src]
/               2      /     2            \       \  sin(x)
\-sin(x) + 2*cos (x) - \- cos (x) + sin(x)/*sin(x)/*e      
((sin(x)cos2(x))sin(x)sin(x)+2cos2(x))esin(x)\left(- \left(\sin{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} - \sin{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right) e^{\sin{\left(x \right)}}
The third derivative [src]
 /         2                 /       2              \       \         sin(x)
-\1 - 3*cos (x) + 6*sin(x) + \1 - cos (x) + 3*sin(x)/*sin(x)/*cos(x)*e      
((3sin(x)cos2(x)+1)sin(x)+6sin(x)3cos2(x)+1)esin(x)cos(x)- \left(\left(3 \sin{\left(x \right)} - \cos^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + 6 \sin{\left(x \right)} - 3 \cos^{2}{\left(x \right)} + 1\right) e^{\sin{\left(x \right)}} \cos{\left(x \right)}
The graph
Derivative of е^sinx*sinx