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The derivative of the function/ sin(x^2-1)

Derivative of sin(x^2-1)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
   / 2    \
sin\x  - 1/
sin⁡(x2−1)\sin{\left(x^{2} - 1 \right)} 
sin(x^2 - 1)
Detail solution

  1. Let u=x2−1u = x^{2} - 1.

  2. The derivative of sine is cosine:

    ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

  3. Then, apply the chain rule. Multiply by ddx(x2−1)\frac{d}{d x} \left(x^{2} - 1\right):

    1. Differentiate x2−1x^{2} - 1 term by term:

      1. Apply the power rule: x2x^{2} goes to 2x2 x

      2. The derivative of the constant −1-1 is zero.

      The result is: 2x2 x

    The result of the chain rule is:

    2xcos⁡(x2−1)2 x \cos{\left(x^{2} - 1 \right)}

  4. Now simplify:

    2xcos⁡(x2−1)2 x \cos{\left(x^{2} - 1 \right)}

The answer is:

2xcos⁡(x2−1)2 x \cos{\left(x^{2} - 1 \right)}

The graph
02468-8-6-4-2-1010-5050
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
       / 2    \
2*x*cos\x  - 1/
2xcos⁡(x2−1)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(−2x2sin⁡(x2−1)+cos⁡(x2−1))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(4x4sin⁡(x2−1)−12x2cos⁡(x2−1)−3sin⁡(x2−1))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 //
−4x(2x2cos⁡(x2−1)+3sin⁡(x2−1))- 4 x \left(2 x^{2} \cos{\left(x^{2} - 1 \right)} + 3 \sin{\left(x^{2} - 1 \right)}\right)
Simplify
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