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Derivative of cos(x^2+1)

You have entered [src]

   / 2    \
cos\x  + 1/

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

d /   / 2    \\
--\cos\x  + 1//
dx

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

Transform

Detail solution

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph