Derivative of sin(2x^2+1)

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   /   2    \
sin\2*x  + 1/

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

d /   /   2    \\
--\sin\2*x  + 1//
dx

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

Transform

Detail solution

Let $u = 2 x^{2} + 1$ .
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 $\frac{d}{d x} \left(2 x^{2} + 1\right)$ :
1. Differentiate $2 x^{2} + 1$ 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: $x^{2}$ goes to $2 x$
    So, the result is: $4 x$
  2. The derivative of the constant $1$ is zero.
  The result is: $4 x$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

      /     /       2\      2    /       2\\
-16*x*\3*sin\1 + 2*x / + 4*x *cos\1 + 2*x //

$$- 16 x \left(4 x^{2} \cos{\left(2 x^{2} + 1 \right)} + 3 \sin{\left(2 x^{2} + 1 \right)}\right)$$

Simplify

The graph