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The derivative of the function/ ax*sinx

Derivative of ax*sinx

The solution

You have entered [src]

a*x*sin(x)

$$a x \sin{\left(x \right)}$$

d             
--(a*x*sin(x))
dx

$$\frac{\partial}{\partial x} a x \sin{\left(x \right)}$$

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:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $x \cos{\left(x \right)} + \sin{\left(x \right)}$
So, the result is: $a \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)$

The answer is:

$a \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)$

The first derivative [src]

a*sin(x) + a*x*cos(x)

$$a x \cos{\left(x \right)} + a \sin{\left(x \right)}$$

Simplify

The second derivative [src]

a*(2*cos(x) - x*sin(x))

$$a \left(- x \sin{\left(x \right)} + 2 \cos{\left(x \right)}\right)$$

Simplify

The third derivative [src]

-a*(3*sin(x) + x*cos(x))

$$- a \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right)$$

Simplify