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Identical expressions
sqrtcos(3x^ two)
square root of co sinus of e of (3x squared )
square root of co sinus of e of (3x to the power of two)
√cos(3x^2)
sqrtcos(3x2)
sqrtcos3x2
sqrtcos(3x²)
sqrtcos(3x to the power of 2)
sqrtcos3x^2

The derivative of the function/ sqrtcos(3x^2)

Derivative of sqrtcos(3x^2)

The solution

You have entered [src]

   ___________
  /    /   2\ 
\/  cos\3*x /

$$\sqrt{\cos{\left(3 x^{2} \right)}}$$

sqrt(cos(3*x^2))

Detail solution

Let $u = \cos{\left(3 x^{2} \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(3 x^{2} \right)}$ :
1. Let $u = 3 x^{2}$ .
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 $\frac{d}{d x} 3 x^{2}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $6 x$
  The result of the chain rule is:
  
  $- 6 x \sin{\left(3 x^{2} \right)}$
The result of the chain rule is:

$- \frac{3 x \sin{\left(3 x^{2} \right)}}{\sqrt{\cos{\left(3 x^{2} \right)}}}$

The answer is:

$- \frac{3 x \sin{\left(3 x^{2} \right)}}{\sqrt{\cos{\left(3 x^{2} \right)}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        /   2\
-3*x*sin\3*x /
--------------
   ___________
  /    /   2\ 
\/  cos\3*x /

$$- \frac{3 x \sin{\left(3 x^{2} \right)}}{\sqrt{\cos{\left(3 x^{2} \right)}}}$$

Simplify

The second derivative [src]

   /     /   2\              ___________      2    2/   2\\
   |  sin\3*x /         2   /    /   2\    3*x *sin \3*x /|
-3*|-------------- + 6*x *\/  cos\3*x /  + ---------------|
   |   ___________                              3/2/   2\ |
   |  /    /   2\                            cos   \3*x / |
   \\/  cos\3*x /                                         /

$$- 3 \left(\frac{3 x^{2} \sin^{2}{\left(3 x^{2} \right)}}{\cos^{\frac{3}{2}}{\left(3 x^{2} \right)}} + 6 x^{2} \sqrt{\cos{\left(3 x^{2} \right)}} + \frac{\sin{\left(3 x^{2} \right)}}{\sqrt{\cos{\left(3 x^{2} \right)}}}\right)$$

Simplify

The third derivative [src]

      /     ___________       2/   2\       2    /   2\      2    3/   2\\
      |    /    /   2\     sin \3*x /    2*x *sin\3*x /   3*x *sin \3*x /|
-27*x*|2*\/  cos\3*x /  + ------------ + -------------- + ---------------|
      |                      3/2/   2\      ___________        5/2/   2\ |
      |                   cos   \3*x /     /    /   2\      cos   \3*x / |
      \                                  \/  cos\3*x /                   /

$$- 27 x \left(\frac{3 x^{2} \sin^{3}{\left(3 x^{2} \right)}}{\cos^{\frac{5}{2}}{\left(3 x^{2} \right)}} + \frac{2 x^{2} \sin{\left(3 x^{2} \right)}}{\sqrt{\cos{\left(3 x^{2} \right)}}} + \frac{\sin^{2}{\left(3 x^{2} \right)}}{\cos^{\frac{3}{2}}{\left(3 x^{2} \right)}} + 2 \sqrt{\cos{\left(3 x^{2} \right)}}\right)$$

Simplify

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Derivative of sqrtcos(3x^2)

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The solution