Mister Exam

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Derivative of:
Derivative of x^(x^2+1)
Derivative of ln(2x)/x
Derivative of e^(3*x)*cos(2*x)
Derivative of cos^2(x^2)
Identical expressions
cos^ two (x^ two)
co sinus of e of squared (x squared )
co sinus of e of to the power of two (x to the power of two)
cos2(x2)
cos2x2
cos²(x²)
cos to the power of 2(x to the power of 2)
cos^2x^2

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

Derivative of cos^2(x^2)

The solution

You have entered [src]

   2/ 2\
cos \x /

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

cos(x^2)^2

Detail solution

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

$- 4 x \sin{\left(x^{2} \right)} \cos{\left(x^{2} \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        / 2\    / 2\
-4*x*cos\x /*sin\x /

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

Simplify

The second derivative [src]

  /     / 2\    / 2\      2    2/ 2\      2    2/ 2\\
4*\- cos\x /*sin\x / - 2*x *cos \x / + 2*x *sin \x //

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

Simplify

The third derivative [src]

    /       2/ 2\        2/ 2\      2    / 2\    / 2\\
8*x*\- 3*cos \x / + 3*sin \x / + 8*x *cos\x /*sin\x //

$$8 x \left(8 x^{2} \sin{\left(x^{2} \right)} \cos{\left(x^{2} \right)} + 3 \sin^{2}{\left(x^{2} \right)} - 3 \cos^{2}{\left(x^{2} \right)}\right)$$

Simplify

The graph

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Identical expressions

Derivative of cos^2(x^2)

The graph:

Piecewise:

The solution