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Derivative of:
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Derivative of sin^2(x^2)
Derivative of log(1+1/(x^2))
Identical expressions
sin^ two (x^ two)
sinus of squared (x squared )
sinus of to the power of two (x to the power of two)
sin2(x2)
sin2x2
sin²(x²)
sin to the power of 2(x to the power of 2)
sin^2x^2

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

Derivative of sin^2(x^2)

The solution

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   2/ 2\
sin \x /

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

sin(x^2)^2

Detail solution

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

The graph:

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The solution