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sin³x²
The derivative of the function/ sin³x²

Derivative of sin³x²

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

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The solution

You have entered [src] 
        / 2\
        \3 /
(sin(x))
sin⁡32(x)\sin^{3^{2}}{\left(x \right)} 
  /        / 2\\
d |        \3 /|
--\(sin(x))    /
dx
ddxsin⁡32(x)\frac{d}{d x} \sin^{3^{2}}{\left(x \right)}
Transform
Detail solution

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

  2. Apply the power rule: u32u^{3^{2}} goes to 9u32u\frac{9 u^{3^{2}}}{u}

  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:

    9sin⁡32(x)cos⁡(x)sin⁡(x)\frac{9 \sin^{3^{2}}{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}

  4. Now simplify:

    9sin⁡8(x)cos⁡(x)9 \sin^{8}{\left(x \right)} \cos{\left(x \right)}

The answer is:

9sin⁡8(x)cos⁡(x)9 \sin^{8}{\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] 
          / 2\       
          \3 /       
9*(sin(x))    *cos(x)
---------------------
        sin(x)
9sin⁡32(x)cos⁡(x)sin⁡(x)\frac{9 \sin^{3^{2}}{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}
Simplify
The second derivative [src] 
     7    /     2           2   \
9*sin (x)*\- sin (x) + 8*cos (x)/
9(−sin⁡2(x)+8cos⁡2(x))sin⁡7(x)9 \left(- \sin^{2}{\left(x \right)} + 8 \cos^{2}{\left(x \right)}\right) \sin^{7}{\left(x \right)}
Simplify
The third derivative [src] 
     6    /        2            2   \       
9*sin (x)*\- 25*sin (x) + 56*cos (x)/*cos(x)
9(−25sin⁡2(x)+56cos⁡2(x))sin⁡6(x)cos⁡(x)9 \left(- 25 \sin^{2}{\left(x \right)} + 56 \cos^{2}{\left(x \right)}\right) \sin^{6}{\left(x \right)} \cos{\left(x \right)}
Simplify
The graph
Derivative of sin³x²
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