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Derivative of:
Derivative of (x+x^2)^x
Derivative of ln(x^2-2x)
Derivative of 6/x^4
Derivative of 5/x^3
Graphing y =:
2(cos(t))^3
Identical expressions
two (cos(t))^ three
2( co sinus of e of (t)) cubed
two ( co sinus of e of (t)) to the power of three
2(cos(t))3
2cost3
2(cos(t))³
2(cos(t)) to the power of 3
2cost^3

The derivative of the function/ 2(cos(t))^3

Derivative of 2(cos(t))^3

The solution

You have entered [src]

     3   
2*cos (t)

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

2*cos(t)^3

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \cos{\left(t \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d t} \cos{\left(t \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}$
  The result of the chain rule is:
  
  $- 3 \sin{\left(t \right)} \cos^{2}{\left(t \right)}$
So, the result is: $- 6 \sin{\left(t \right)} \cos^{2}{\left(t \right)}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      2          
-6*cos (t)*sin(t)

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

Simplify

The second derivative [src]

  /     2           2   \       
6*\- cos (t) + 2*sin (t)/*cos(t)

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

Simplify

The third derivative [src]

   /       2           2   \       
-6*\- 7*cos (t) + 2*sin (t)/*sin(t)

$$- 6 \left(2 \sin^{2}{\left(t \right)} - 7 \cos^{2}{\left(t \right)}\right) \sin{\left(t \right)}$$

Simplify