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y=sin(2x^2-3)

Derivative of y=sin(2x^2-3)

Function f() - derivative -N order at the point
v

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The solution

You have entered [src]
   /   2    \
sin\2*x  - 3/
sin(2x23)\sin{\left(2 x^{2} - 3 \right)}
d /   /   2    \\
--\sin\2*x  - 3//
dx               
ddxsin(2x23)\frac{d}{d x} \sin{\left(2 x^{2} - 3 \right)}
Detail solution
  1. Let u=2x23u = 2 x^{2} - 3.

  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(2x23)\frac{d}{d x} \left(2 x^{2} - 3\right):

    1. Differentiate 2x232 x^{2} - 3 term by term:

      1. The derivative of a constant times a function is the constant times the derivative of the function.

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

        So, the result is: 4x4 x

      2. The derivative of the constant (1)3\left(-1\right) 3 is zero.

      The result is: 4x4 x

    The result of the chain rule is:

    4xcos(2x23)4 x \cos{\left(2 x^{2} - 3 \right)}

  4. Now simplify:

    4xcos(2x23)4 x \cos{\left(2 x^{2} - 3 \right)}


The answer is:

4xcos(2x23)4 x \cos{\left(2 x^{2} - 3 \right)}

The graph
02468-8-6-4-2-1010-100100
The first derivative [src]
       /   2    \
4*x*cos\2*x  - 3/
4xcos(2x23)4 x \cos{\left(2 x^{2} - 3 \right)}
The second derivative [src]
  /     2    /        2\      /        2\\
4*\- 4*x *sin\-3 + 2*x / + cos\-3 + 2*x //
4(4x2sin(2x23)+cos(2x23))4 \left(- 4 x^{2} \sin{\left(2 x^{2} - 3 \right)} + \cos{\left(2 x^{2} - 3 \right)}\right)
The third derivative [src]
      /     /        2\      2    /        2\\
-16*x*\3*sin\-3 + 2*x / + 4*x *cos\-3 + 2*x //
16x(4x2cos(2x23)+3sin(2x23))- 16 x \left(4 x^{2} \cos{\left(2 x^{2} - 3 \right)} + 3 \sin{\left(2 x^{2} - 3 \right)}\right)
The graph
Derivative of y=sin(2x^2-3)