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Derivative of:
Derivative of 2*x^5
Derivative of 2x²
Derivative of 5-7x
Derivative of e^x*x^2
Identical expressions
y=x^ five *sin3x^ seven
y equally x to the power of 5 multiply by sinus of 3x to the power of 7
y equally x to the power of five multiply by sinus of 3x to the power of seven
y=x5*sin3x7
y=x⁵*sin3x⁷
y=x^5sin3x^7
y=x5sin3x7

The derivative of the function/ y=x^5*sin3x^7

Derivative of y=x^5*sin3x^7

The solution

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 5    7     
x *sin (3*x)

$$x^{5} \sin^{7}{\left(3 x \right)}$$

d / 5    7     \
--\x *sin (3*x)/
dx

$$\frac{d}{d x} x^{5} \sin^{7}{\left(3 x \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{5}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$
$g{\left(x \right)} = \sin^{7}{\left(3 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sin{\left(3 x \right)}$ .
2. Apply the power rule: $u^{7}$ goes to $7 u^{6}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(3 x \right)}$ :
  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 of the chain rule is:
  
  $21 \sin^{6}{\left(3 x \right)} \cos{\left(3 x \right)}$
The result is: $21 x^{5} \sin^{6}{\left(3 x \right)} \cos{\left(3 x \right)} + 5 x^{4} \sin^{7}{\left(3 x \right)}$
Now simplify:

$x^{4} \cdot \left(21 x \cos{\left(3 x \right)} + 5 \sin{\left(3 x \right)}\right) \sin^{6}{\left(3 x \right)}$

The answer is:

$x^{4} \cdot \left(21 x \cos{\left(3 x \right)} + 5 \sin{\left(3 x \right)}\right) \sin^{6}{\left(3 x \right)}$

The graph

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Plot the graph f'(x)

The first derivative [src]

   4    7            5    6              
5*x *sin (3*x) + 21*x *sin (3*x)*cos(3*x)

$$21 x^{5} \sin^{6}{\left(3 x \right)} \cos{\left(3 x \right)} + 5 x^{4} \sin^{7}{\left(3 x \right)}$$

Simplify

The second derivative [src]

 3    5      /      2            2 /   2             2     \                          \
x *sin (3*x)*\20*sin (3*x) - 63*x *\sin (3*x) - 6*cos (3*x)/ + 210*x*cos(3*x)*sin(3*x)/

$$x^{3} \left(- 63 x^{2} \left(\sin^{2}{\left(3 x \right)} - 6 \cos^{2}{\left(3 x \right)}\right) + 210 x \sin{\left(3 x \right)} \cos{\left(3 x \right)} + 20 \sin^{2}{\left(3 x \right)}\right) \sin^{5}{\left(3 x \right)}$$

Simplify

The third derivative [src]

   2    4      /      3             2 /   2             2     \                3 /        2              2     \                     2              \
3*x *sin (3*x)*\20*sin (3*x) - 315*x *\sin (3*x) - 6*cos (3*x)/*sin(3*x) - 63*x *\- 30*cos (3*x) + 19*sin (3*x)/*cos(3*x) + 420*x*sin (3*x)*cos(3*x)/

$$3 x^{2} \left(- 63 x^{3} \cdot \left(19 \sin^{2}{\left(3 x \right)} - 30 \cos^{2}{\left(3 x \right)}\right) \cos{\left(3 x \right)} - 315 x^{2} \left(\sin^{2}{\left(3 x \right)} - 6 \cos^{2}{\left(3 x \right)}\right) \sin{\left(3 x \right)} + 420 x \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)} + 20 \sin^{3}{\left(3 x \right)}\right) \sin^{4}{\left(3 x \right)}$$

Simplify

The graph

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Derivative of y=x^5*sin3x^7

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The solution