Derivative of sin(x²+3)

You have entered [src]

   / 2    \
sin\x  + 3/

$$\sin{\left(x^{2} + 3 \right)}$$

sin(x^2 + 3)

Detail solution

Let $u = x^{2} + 3$ .
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 $\frac{d}{d x} \left(x^{2} + 3\right)$ :
1. Differentiate $x^{2} + 3$ term by term:
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  2. The derivative of the constant $3$ is zero.
  The result is: $2 x$
The result of the chain rule is:

$2 x \cos{\left(x^{2} + 3 \right)}$
Now simplify:

$2 x \cos{\left(x^{2} + 3 \right)}$

The answer is:

$2 x \cos{\left(x^{2} + 3 \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       / 2    \
2*x*cos\x  + 3/

$$2 x \cos{\left(x^{2} + 3 \right)}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify