Derivative of cos(3t^2)

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   /   2\
cos\3*t /

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

cos(3*t^2)

Detail solution

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

$- 6 t \sin{\left(3 t^{2} \right)}$

The answer is:

$- 6 t \sin{\left(3 t^{2} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        /   2\
-6*t*sin\3*t /

$$- 6 t \sin{\left(3 t^{2} \right)}$$

Simplify

The second derivative [src]

   /   2    /   2\      /   2\\
-6*\6*t *cos\3*t / + sin\3*t //

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

Simplify

The third derivative [src]

      /     /   2\      2    /   2\\
108*t*\- cos\3*t / + 2*t *sin\3*t //

$$108 t \left(2 t^{2} \sin{\left(3 t^{2} \right)} - \cos{\left(3 t^{2} \right)}\right)$$

Simplify