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Derivative of:
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Identical expressions
five *sin(x)*sin(three *x)
5 multiply by sinus of (x) multiply by sinus of (3 multiply by x)
five multiply by sinus of (x) multiply by sinus of (three multiply by x)
5sin(x)sin(3x)
5sinxsin3x
Similar expressions
5*sinx*sin(3*x)

The derivative of the function/ 5*sin(x)*sin(3*x)

Derivative of 5sin(x)sin(3*x)

The solution

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5*sin(x)*sin(3*x)

$$5 \sin{\left(x \right)} \sin{\left(3 x \right)}$$

(5*sin(x))*sin(3*x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 5 \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  So, the result is: $5 \cos{\left(x \right)}$
$g{\left(x \right)} = \sin{\left(3 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 3 x$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $3$
  The result of the chain rule is:
  
  $3 \cos{\left(3 x \right)}$
The result is: $15 \sin{\left(x \right)} \cos{\left(3 x \right)} + 5 \sin{\left(3 x \right)} \cos{\left(x \right)}$
Now simplify:

$- 5 \sin{\left(2 x \right)} + 10 \sin{\left(4 x \right)}$

The answer is:

$- 5 \sin{\left(2 x \right)} + 10 \sin{\left(4 x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

5*cos(x)*sin(3*x) + 15*cos(3*x)*sin(x)

$$15 \sin{\left(x \right)} \cos{\left(3 x \right)} + 5 \sin{\left(3 x \right)} \cos{\left(x \right)}$$

Simplify

The second derivative [src]

10*(-5*sin(x)*sin(3*x) + 3*cos(x)*cos(3*x))

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

Simplify

The third derivative [src]

-20*(7*cos(x)*sin(3*x) + 9*cos(3*x)*sin(x))

$$- 20 \left(9 \sin{\left(x \right)} \cos{\left(3 x \right)} + 7 \sin{\left(3 x \right)} \cos{\left(x \right)}\right)$$

Simplify