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Similar expressions
y=sin^2(x^2-1)

The derivative of the function/ y=sin^2(x^2+1)

Derivative of y=sin^2(x^2+1)

The solution

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   2/ 2    \
sin \x  + 1/

$$\sin^{2}{\left(x^{2} + 1 \right)}$$

sin(x^2 + 1)^2

Detail solution

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

$4 x \sin{\left(x^{2} + 1 \right)} \cos{\left(x^{2} + 1 \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       / 2    \    / 2    \
4*x*cos\x  + 1/*sin\x  + 1/

$$4 x \sin{\left(x^{2} + 1 \right)} \cos{\left(x^{2} + 1 \right)}$$

Simplify

The second derivative [src]

  /   /     2\    /     2\      2    2/     2\      2    2/     2\\
4*\cos\1 + x /*sin\1 + x / - 2*x *sin \1 + x / + 2*x *cos \1 + x //

$$4 \left(- 2 x^{2} \sin^{2}{\left(x^{2} + 1 \right)} + 2 x^{2} \cos^{2}{\left(x^{2} + 1 \right)} + \sin{\left(x^{2} + 1 \right)} \cos{\left(x^{2} + 1 \right)}\right)$$

Simplify

The third derivative [src]

    /       2/     2\        2/     2\      2    /     2\    /     2\\
8*x*\- 3*sin \1 + x / + 3*cos \1 + x / - 8*x *cos\1 + x /*sin\1 + x //

$$8 x \left(- 8 x^{2} \sin{\left(x^{2} + 1 \right)} \cos{\left(x^{2} + 1 \right)} - 3 \sin^{2}{\left(x^{2} + 1 \right)} + 3 \cos^{2}{\left(x^{2} + 1 \right)}\right)$$

Simplify

The graph

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Derivative of y=sin^2(x^2+1)

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The solution