Mister Exam

Derivative of sin^4

Function f() - derivative -N order at the point
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The graph:

from to

Piecewise:

The solution

You have entered [src]
   4   
sin (x)
sin4(x)\sin^{4}{\left(x \right)}
sin(x)^4
Detail solution
  1. Let u=sin(x)u = \sin{\left(x \right)}.

  2. Apply the power rule: u4u^{4} goes to 4u34 u^{3}

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

    1. The derivative of sine is cosine:

      ddxsin(x)=cos(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

    The result of the chain rule is:

    4sin3(x)cos(x)4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}


The answer is:

4sin3(x)cos(x)4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}

The graph
02468-8-6-4-2-10102.5-2.5
The first derivative [src]
     3          
4*sin (x)*cos(x)
4sin3(x)cos(x)4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}
The second derivative [src]
     2    /     2           2   \
4*sin (x)*\- sin (x) + 3*cos (x)/
4(sin2(x)+3cos2(x))sin2(x)4 \left(- \sin^{2}{\left(x \right)} + 3 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)}
The third derivative [src]
  /       2           2   \              
8*\- 5*sin (x) + 3*cos (x)/*cos(x)*sin(x)
8(5sin2(x)+3cos2(x))sin(x)cos(x)8 \left(- 5 \sin^{2}{\left(x \right)} + 3 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}
The graph
Derivative of sin^4