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The derivative of the function/ А*(x/3)*sin(x/3)

Derivative of А*(x/3)*sin(x/3)

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

You have entered [src] 
  x    /x\
a*-*sin|-|
  3    \3/
ax3sin⁡(x3)a \frac{x}{3} \sin{\left(\frac{x}{3} \right)} 
(a*(x/3))*sin(x/3)
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

    f(x)=axsin⁡(x3)f{\left(x \right)} = a x \sin{\left(\frac{x}{3} \right)} and g(x)=3g{\left(x \right)} = 3.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Apply the product rule:

        ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

        f(x)=xf{\left(x \right)} = x; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. Apply the power rule: xx goes to 11

        g(x)=sin⁡(x3)g{\left(x \right)} = \sin{\left(\frac{x}{3} \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. Let u=x3u = \frac{x}{3}.

        2. The derivative of sine is cosine:

          ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

        3. Then, apply the chain rule. Multiply by ddxx3\frac{d}{d x} \frac{x}{3}:

          1. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: xx goes to 11

            So, the result is: 13\frac{1}{3}

          The result of the chain rule is:

          cos⁡(x3)3\frac{\cos{\left(\frac{x}{3} \right)}}{3}

        The result is: xcos⁡(x3)3+sin⁡(x3)\frac{x \cos{\left(\frac{x}{3} \right)}}{3} + \sin{\left(\frac{x}{3} \right)}

      So, the result is: a(xcos⁡(x3)3+sin⁡(x3))a \left(\frac{x \cos{\left(\frac{x}{3} \right)}}{3} + \sin{\left(\frac{x}{3} \right)}\right)

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. The derivative of the constant 33 is zero.

    Now plug in to the quotient rule:

    a(xcos⁡(x3)3+sin⁡(x3))3\frac{a \left(\frac{x \cos{\left(\frac{x}{3} \right)}}{3} + \sin{\left(\frac{x}{3} \right)}\right)}{3}

  2. Now simplify:

    a(xcos⁡(x3)+3sin⁡(x3))9\frac{a \left(x \cos{\left(\frac{x}{3} \right)} + 3 \sin{\left(\frac{x}{3} \right)}\right)}{9}

The answer is:

a(xcos⁡(x3)+3sin⁡(x3))9\frac{a \left(x \cos{\left(\frac{x}{3} \right)} + 3 \sin{\left(\frac{x}{3} \right)}\right)}{9}

The first derivative [src] 
     /x\          /x\
a*sin|-|   a*x*cos|-|
     \3/          \3/
-------- + ----------
   3           9
axcos⁡(x3)9+asin⁡(x3)3\frac{a x \cos{\left(\frac{x}{3} \right)}}{9} + \frac{a \sin{\left(\frac{x}{3} \right)}}{3}
Simplify
The second derivative [src] 
  /     /x\        /x\\
a*|6*cos|-| - x*sin|-||
  \     \3/        \3//
-----------------------
           27
a(−xsin⁡(x3)+6cos⁡(x3))27\frac{a \left(- x \sin{\left(\frac{x}{3} \right)} + 6 \cos{\left(\frac{x}{3} \right)}\right)}{27}
Simplify
The third derivative [src] 
   /     /x\        /x\\ 
-a*|9*sin|-| + x*cos|-|| 
   \     \3/        \3// 
-------------------------
            81
−a(xcos⁡(x3)+9sin⁡(x3))81- \frac{a \left(x \cos{\left(\frac{x}{3} \right)} + 9 \sin{\left(\frac{x}{3} \right)}\right)}{81}
Simplify
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