Derivative of y=sin(2x^3)

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   /   3\
sin\2*x /

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

d /   /   3\\
--\sin\2*x //
dx

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

Transform

Detail solution

Let $u = 2 x^{3}$ .
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 $\frac{d}{d x} 2 x^{3}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
  So, the result is: $6 x^{2}$
The result of the chain rule is:

$6 x^{2} \cos{\left(2 x^{3} \right)}$

The answer is:

$6 x^{2} \cos{\left(2 x^{3} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2    /   3\
6*x *cos\2*x /

$$6 x^{2} \cos{\left(2 x^{3} \right)}$$

Simplify

The second derivative [src]

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

$$12 x \left(- 3 x^{3} \sin{\left(2 x^{3} \right)} + \cos{\left(2 x^{3} \right)}\right)$$

Simplify

The third derivative [src]

   /      3    /   3\       6    /   3\      /   3\\
12*\- 18*x *sin\2*x / - 18*x *cos\2*x / + cos\2*x //

$$12 \left(- 18 x^{6} \cos{\left(2 x^{3} \right)} - 18 x^{3} \sin{\left(2 x^{3} \right)} + \cos{\left(2 x^{3} \right)}\right)$$

Simplify

The graph