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(e^(cosx)+2)^2
The derivative of the function/ (e^(cosx)+2)^2

Derivative of (e^(cosx)+2)^2

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

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The solution

You have entered [src] 
             2
/ cos(x)    \ 
\e       + 2/
(ecos⁡(x)+2)2\left(e^{\cos{\left(x \right)}} + 2\right)^{2} 
  /             2\
d |/ cos(x)    \ |
--\\e       + 2/ /
dx
ddx(ecos⁡(x)+2)2\frac{d}{d x} \left(e^{\cos{\left(x \right)}} + 2\right)^{2}
Transform
Detail solution

  1. Let u=ecos⁡(x)+2u = e^{\cos{\left(x \right)}} + 2.

  2. Apply the power rule: u2u^{2} goes to 2u2 u

  3. Then, apply the chain rule. Multiply by ddx(ecos⁡(x)+2)\frac{d}{d x} \left(e^{\cos{\left(x \right)}} + 2\right):

    1. Differentiate ecos⁡(x)+2e^{\cos{\left(x \right)}} + 2 term by term:

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

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

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

        1. The derivative of cosine is negative sine:

          ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

        The result of the chain rule is:

        −ecos⁡(x)sin⁡(x)- e^{\cos{\left(x \right)}} \sin{\left(x \right)}

      4. The derivative of the constant 22 is zero.

      The result is: −ecos⁡(x)sin⁡(x)- e^{\cos{\left(x \right)}} \sin{\left(x \right)}

    The result of the chain rule is:

    −(2ecos⁡(x)+4)ecos⁡(x)sin⁡(x)- \left(2 e^{\cos{\left(x \right)}} + 4\right) e^{\cos{\left(x \right)}} \sin{\left(x \right)}

  4. Now simplify:

    −2(ecos⁡(x)+2)ecos⁡(x)sin⁡(x)- 2 \left(e^{\cos{\left(x \right)}} + 2\right) e^{\cos{\left(x \right)}} \sin{\left(x \right)}

The answer is:

−2(ecos⁡(x)+2)ecos⁡(x)sin⁡(x)- 2 \left(e^{\cos{\left(x \right)}} + 2\right) e^{\cos{\left(x \right)}} \sin{\left(x \right)}

The graph
02468-8-6-4-2-1010-5050
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   / cos(x)    \  cos(x)       
-2*\e       + 2/*e      *sin(x)
−2(ecos⁡(x)+2)ecos⁡(x)sin⁡(x)- 2 \left(e^{\cos{\left(x \right)}} + 2\right) e^{\cos{\left(x \right)}} \sin{\left(x \right)}
Simplify
The second derivative [src] 
  /   2    /     cos(x)\      2     cos(x)   /     cos(x)\       \  cos(x)
2*\sin (x)*\2 + e      / + sin (x)*e       - \2 + e      /*cos(x)/*e
2((ecos⁡(x)+2)sin⁡2(x)+ecos⁡(x)sin⁡2(x)−(ecos⁡(x)+2)cos⁡(x))ecos⁡(x)2 \left(\left(e^{\cos{\left(x \right)}} + 2\right) \sin^{2}{\left(x \right)} + e^{\cos{\left(x \right)}} \sin^{2}{\left(x \right)} - \left(e^{\cos{\left(x \right)}} + 2\right) \cos{\left(x \right)}\right) e^{\cos{\left(x \right)}}
Simplify
The third derivative [src] 
  /       2    /     cos(x)\        2     cos(x)     /     cos(x)\                    cos(x)    cos(x)\  cos(x)       
2*\2 - sin (x)*\2 + e      / - 3*sin (x)*e       + 3*\2 + e      /*cos(x) + 3*cos(x)*e       + e      /*e      *sin(x)
2(−(ecos⁡(x)+2)sin⁡2(x)−3ecos⁡(x)sin⁡2(x)+3(ecos⁡(x)+2)cos⁡(x)+3ecos⁡(x)cos⁡(x)+ecos⁡(x)+2)ecos⁡(x)sin⁡(x)2 \left(- \left(e^{\cos{\left(x \right)}} + 2\right) \sin^{2}{\left(x \right)} - 3 e^{\cos{\left(x \right)}} \sin^{2}{\left(x \right)} + 3 \left(e^{\cos{\left(x \right)}} + 2\right) \cos{\left(x \right)} + 3 e^{\cos{\left(x \right)}} \cos{\left(x \right)} + e^{\cos{\left(x \right)}} + 2\right) e^{\cos{\left(x \right)}} \sin{\left(x \right)}
Simplify
The graph
Derivative of (e^(cosx)+2)^2
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