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Derivative of:
Derivative of 6^x
Derivative of 2/sqrt(x)
Derivative of (x^2-1)/(x^2+1)
Derivative of x*e^(1/x)
Identical expressions
cossin^ two (x)
co sinus of e of sinus of squared (x)
co sinus of e of sinus of to the power of two (x)
cossin2(x)
cossin2x
cossin²(x)
cossin to the power of 2(x)
cossin^2x
Similar expressions
y=sin(cos^2x)*cos(sin^2x)

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

Derivative of cossin^2(x)

The solution

You have entered [src]

   2        
cos (sin(x))

$$\cos^{2}{\left(\sin{\left(x \right)} \right)}$$

d /   2        \
--\cos (sin(x))/
dx

$$\frac{d}{d x} \cos^{2}{\left(\sin{\left(x \right)} \right)}$$

Transform

Detail solution

Let $u = \cos{\left(\sin{\left(x \right)} \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(\sin{\left(x \right)} \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $- \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}$
The result of the chain rule is:

$- 2 \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-2*cos(x)*cos(sin(x))*sin(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))/

$$2 \left(\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin^{2}{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} \cos^{2}{\left(\sin{\left(x \right)} \right)}\right)$$

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)

$$2 \left(- 3 \sin{\left(x \right)} \sin^{2}{\left(\sin{\left(x \right)} \right)} + 3 \sin{\left(x \right)} \cos^{2}{\left(\sin{\left(x \right)} \right)} + 4 \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)}$$

Simplify

The graph

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Derivative of cossin^2(x)

The graph:

Piecewise:

The solution