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cossin^2(x)
The derivative of the function/ cossin^2(x)

Derivative of cossin^2(x)

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

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The solution

You have entered [src] 
   2        
cos (sin(x))
cos⁡2(sin⁡(x))\cos^{2}{\left(\sin{\left(x \right)} \right)} 
d /   2        \
--\cos (sin(x))/
dx
ddxcos⁡2(sin⁡(x))\frac{d}{d x} \cos^{2}{\left(\sin{\left(x \right)} \right)}
Transform
Detail solution

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

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

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

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

    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 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:

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

    The result of the chain rule is:

    −2sin⁡(sin⁡(x))cos⁡(x)cos⁡(sin⁡(x))- 2 \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}

  4. Now simplify:

    sin⁡(x−2sin⁡(x))2−sin⁡(x+2sin⁡(x))2\frac{\sin{\left(x - 2 \sin{\left(x \right)} \right)}}{2} - \frac{\sin{\left(x + 2 \sin{\left(x \right)} \right)}}{2}

The answer is:

sin⁡(x−2sin⁡(x))2−sin⁡(x+2sin⁡(x))2\frac{\sin{\left(x - 2 \sin{\left(x \right)} \right)}}{2} - \frac{\sin{\left(x + 2 \sin{\left(x \right)} \right)}}{2}

The graph
02468-8-6-4-2-10102-2
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
-2*cos(x)*cos(sin(x))*sin(sin(x))
−2sin⁡(sin⁡(x))cos⁡(x)cos⁡(sin⁡(x))- 2 \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}
Simplify
The second derivative [src] 
  /   2       2              2       2                                         \
2*\cos (x)*sin (sin(x)) - cos (x)*cos (sin(x)) + cos(sin(x))*sin(x)*sin(sin(x))/
2(sin⁡(x)sin⁡(sin⁡(x))cos⁡(sin⁡(x))+sin⁡2(sin⁡(x))cos⁡2(x)−cos⁡2(x)cos⁡2(sin⁡(x)))2 \left(\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin^{2}{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} \cos^{2}{\left(\sin{\left(x \right)} \right)}\right)
Simplify
The third derivative [src] 
  /                               2                       2                       2                           \       
2*\cos(sin(x))*sin(sin(x)) - 3*sin (sin(x))*sin(x) + 3*cos (sin(x))*sin(x) + 4*cos (x)*cos(sin(x))*sin(sin(x))/*cos(x)
2(−3sin⁡(x)sin⁡2(sin⁡(x))+3sin⁡(x)cos⁡2(sin⁡(x))+4sin⁡(sin⁡(x))cos⁡2(x)cos⁡(sin⁡(x))+sin⁡(sin⁡(x))cos⁡(sin⁡(x)))cos⁡(x)2 \left(- 3 \sin{\left(x \right)} \sin^{2}{\left(\sin{\left(x \right)} \right)} + 3 \sin{\left(x \right)} \cos^{2}{\left(\sin{\left(x \right)} \right)} + 4 \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)}
Simplify
The graph
Derivative of cossin^2(x)
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