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y=√sin²*5x/x³+1
The derivative of the function/ y=√sin²*5x/x³+1

Derivative of y=√sin²*5x/x³+1

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

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The solution

You have entered [src] 
          2      
  ________       
\/ sin(5)  *x    
------------- + 1
       3         
      x
1+x(sin⁡(5))2x31 + \frac{x \left(\sqrt{\sin{\left(5 \right)}}\right)^{2}}{x^{3}} 
((sqrt(sin(5)))^2*x)/x^3 + 1
Detail solution

  1. Differentiate 1+x(sin⁡(5))2x31 + \frac{x \left(\sqrt{\sin{\left(5 \right)}}\right)^{2}}{x^{3}} term by term:

    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)=xsin⁡(5)f{\left(x \right)} = x \sin{\left(5 \right)} and g(x)=x3g{\left(x \right)} = x^{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 power rule: xx goes to 11

        So, the result is: sin⁡(5)\sin{\left(5 \right)}

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

      1. Apply the power rule: x3x^{3} goes to 3x23 x^{2}

      Now plug in to the quotient rule:

      −2sin⁡(5)x3- \frac{2 \sin{\left(5 \right)}}{x^{3}}

    2. The derivative of the constant 11 is zero.

    The result is: −2sin⁡(5)x3- \frac{2 \sin{\left(5 \right)}}{x^{3}}

The answer is:

−2sin⁡(5)x3- \frac{2 \sin{\left(5 \right)}}{x^{3}}

The graph
02468-8-6-4-2-1010-50005000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
-2*sin(5)
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     3   
    x
−2sin⁡(5)x3- \frac{2 \sin{\left(5 \right)}}{x^{3}}
Simplify
The second derivative [src] 
6*sin(5)
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    4   
   x
6sin⁡(5)x4\frac{6 \sin{\left(5 \right)}}{x^{4}}
Simplify
The third derivative [src] 
-24*sin(5)
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     5    
    x
−24sin⁡(5)x5- \frac{24 \sin{\left(5 \right)}}{x^{5}}
Simplify
The graph
Derivative of y=√sin²*5x/x³+1
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