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sin^5x
The derivative of the function/ sin^5x

Derivative of sin^5x

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

The graph:

from to

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The solution

You have entered [src] 
   5   
sin (x)
sin⁡5(x)\sin^{5}{\left(x \right)} 
d /   5   \
--\sin (x)/
dx
ddxsin⁡5(x)\frac{d}{d x} \sin^{5}{\left(x \right)}
Transform
Detail solution

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

  2. Apply the power rule: u5u^{5} goes to 5u45 u^{4}

  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:

    5sin⁡4(x)cos⁡(x)5 \sin^{4}{\left(x \right)} \cos{\left(x \right)}

The answer is:

5sin⁡4(x)cos⁡(x)5 \sin^{4}{\left(x \right)} \cos{\left(x \right)}

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
     4          
5*sin (x)*cos(x)
5sin⁡4(x)cos⁡(x)5 \sin^{4}{\left(x \right)} \cos{\left(x \right)}
Simplify
The second derivative [src] 
     3    /     2           2   \
5*sin (x)*\- sin (x) + 4*cos (x)/
5(−sin⁡2(x)+4cos⁡2(x))sin⁡3(x)5 \left(- \sin^{2}{\left(x \right)} + 4 \cos^{2}{\left(x \right)}\right) \sin^{3}{\left(x \right)}
Simplify
The third derivative [src] 
     2    /        2            2   \       
5*sin (x)*\- 13*sin (x) + 12*cos (x)/*cos(x)
5(−13sin⁡2(x)+12cos⁡2(x))sin⁡2(x)cos⁡(x)5 \left(- 13 \sin^{2}{\left(x \right)} + 12 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)}
Simplify
The graph
Derivative of sin^5x
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