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Derivative of sin(x^2-1)

You have entered [src]

   / 2    \
sin\x  - 1/

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

sin(x^2 - 1)

Detail solution

Let $u = x^{2} - 1$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       / 2    \
2*x*cos\x  - 1/

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

Simplify

The second derivative [src]

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

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

Simplify

4-я производная [src]

  /       /      2\       2    /      2\      4    /      2\\
4*\- 3*sin\-1 + x / - 12*x *cos\-1 + x / + 4*x *sin\-1 + x //

$$4 \left(4 x^{4} \sin{\left(x^{2} - 1 \right)} - 12 x^{2} \cos{\left(x^{2} - 1 \right)} - 3 \sin{\left(x^{2} - 1 \right)}\right)$$

Simplify

The third derivative [src]

     /     /      2\      2    /      2\\
-4*x*\3*sin\-1 + x / + 2*x *cos\-1 + x //

$$- 4 x \left(2 x^{2} \cos{\left(x^{2} - 1 \right)} + 3 \sin{\left(x^{2} - 1 \right)}\right)$$

Simplify