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Derivative of y=5x^5*sinx

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   5       
5*x *sin(x)

$$5 x^{5} \sin{\left(x \right)}$$

d /   5       \
--\5*x *sin(x)/
dx

$$\frac{d}{d x} 5 x^{5} \sin{\left(x \right)}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{5}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result is: $x^{5} \cos{\left(x \right)} + 5 x^{4} \sin{\left(x \right)}$
So, the result is: $5 x^{5} \cos{\left(x \right)} + 25 x^{4} \sin{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   5              4       
5*x *cos(x) + 25*x *sin(x)

$$5 x^{5} \cos{\left(x \right)} + 25 x^{4} \sin{\left(x \right)}$$

Simplify

The second derivative [src]

   3 /             2                     \
5*x *\20*sin(x) - x *sin(x) + 10*x*cos(x)/

$$5 x^{3} \left(- x^{2} \sin{\left(x \right)} + 10 x \cos{\left(x \right)} + 20 \sin{\left(x \right)}\right)$$

Simplify

The third derivative [src]

   2 /             3              2                     \
5*x *\60*sin(x) - x *cos(x) - 15*x *sin(x) + 60*x*cos(x)/

$$5 x^{2} \left(- x^{3} \cos{\left(x \right)} - 15 x^{2} \sin{\left(x \right)} + 60 x \cos{\left(x \right)} + 60 \sin{\left(x \right)}\right)$$

Simplify

The graph