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sin^3t
The derivative of the function/ sin^3t

Derivative of sin^3t

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
   3   
sin (t)
sin⁡3(t)\sin^{3}{\left(t \right)} 
d /   3   \
--\sin (t)/
dt
ddtsin⁡3(t)\frac{d}{d t} \sin^{3}{\left(t \right)}
Transform
Detail solution

  1. Let u=sin⁡(t)u = \sin{\left(t \right)}.

  2. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

  3. Then, apply the chain rule. Multiply by ddtsin⁡(t)\frac{d}{d t} \sin{\left(t \right)}:

    1. The derivative of sine is cosine:

      ddtsin⁡(t)=cos⁡(t)\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}

    The result of the chain rule is:

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

The answer is:

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

The graph
02468-8-6-4-2-10102.5-2.5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
     2          
3*sin (t)*cos(t)
3sin⁡2(t)cos⁡(t)3 \sin^{2}{\left(t \right)} \cos{\left(t \right)}
Simplify
The second derivative [src] 
  /     2           2   \       
3*\- sin (t) + 2*cos (t)/*sin(t)
3(−sin⁡2(t)+2cos⁡2(t))sin⁡(t)3 \left(- \sin^{2}{\left(t \right)} + 2 \cos^{2}{\left(t \right)}\right) \sin{\left(t \right)}
Simplify
The third derivative [src] 
  /       2           2   \       
3*\- 7*sin (t) + 2*cos (t)/*cos(t)
3(−7sin⁡2(t)+2cos⁡2(t))cos⁡(t)3 \left(- 7 \sin^{2}{\left(t \right)} + 2 \cos^{2}{\left(t \right)}\right) \cos{\left(t \right)}
Simplify
The graph
Derivative of sin^3t
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