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

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

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   / 2        \
sin\x  + x + 1/
$$\sin{\left(\left(x^{2} + x\right) + 1 \right)}$$
sin(x^2 + x + 1)
Detail solution
  1. Let .

  2. The derivative of sine is cosine:

  3. Then, apply the chain rule. Multiply by :

    1. Differentiate term by term:

      1. Differentiate term by term:

        1. Apply the power rule: goes to

        2. Apply the power rule: goes to

        The result is:

      2. The derivative of the constant is zero.

      The result is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
             / 2        \
(1 + 2*x)*cos\x  + x + 1/
$$\left(2 x + 1\right) \cos{\left(\left(x^{2} + x\right) + 1 \right)}$$
The second derivative [src]
     /         2\            2    /         2\
2*cos\1 + x + x / - (1 + 2*x) *sin\1 + x + x /
$$- \left(2 x + 1\right)^{2} \sin{\left(x^{2} + x + 1 \right)} + 2 \cos{\left(x^{2} + x + 1 \right)}$$
The third derivative [src]
           /     /         2\            2    /         2\\
-(1 + 2*x)*\6*sin\1 + x + x / + (1 + 2*x) *cos\1 + x + x //
$$- \left(2 x + 1\right) \left(\left(2 x + 1\right)^{2} \cos{\left(x^{2} + x + 1 \right)} + 6 \sin{\left(x^{2} + x + 1 \right)}\right)$$
The graph
Derivative of y=sin(x^2+x+1)