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Derivative of:
Derivative of e^x/x^2
Derivative of 5x^3
Derivative of 5*x^2-2/sqrt(x)+sin(pi/4)
Derivative of 2^x+1
Identical expressions
y=sin(2x)^ three *cos8x^ five
y equally sinus of (2x) cubed multiply by co sinus of e of 8x to the power of 5
y equally sinus of (2x) to the power of three multiply by co sinus of e of 8x to the power of five
y=sin(2x)3*cos8x5
y=sin2x3*cos8x5
y=sin(2x)³*cos8x⁵
y=sin(2x) to the power of 3*cos8x to the power of 5
y=sin(2x)^3cos8x^5
y=sin(2x)3cos8x5
y=sin2x3cos8x5
y=sin2x^3cos8x^5

The derivative of the function/ y=sin(2x)^3*cos8x^5

Derivative of y=sin(2x)^3*cos8x^5

The solution

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   3         5     
sin (2*x)*cos (8*x)

$$\sin^{3}{\left(2 x \right)} \cos^{5}{\left(8 x \right)}$$

d /   3         5     \
--\sin (2*x)*cos (8*x)/
dx

$$\frac{d}{d x} \sin^{3}{\left(2 x \right)} \cos^{5}{\left(8 x \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin^{3}{\left(2 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(2 x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(2 x \right)}$ :
  1. Let $u = 2 x$ .
  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 $\frac{d}{d x} 2 x$ :
    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: $2$
    The result of the chain rule is:
    
    $2 \cos{\left(2 x \right)}$
  The result of the chain rule is:
  
  $6 \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)}$
$g{\left(x \right)} = \cos^{5}{\left(8 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \cos{\left(8 x \right)}$ .
2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(8 x \right)}$ :
  1. Let $u = 8 x$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 8 x$ :
    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: $8$
    The result of the chain rule is:
    
    $- 8 \sin{\left(8 x \right)}$
  The result of the chain rule is:
  
  $- 40 \sin{\left(8 x \right)} \cos^{4}{\left(8 x \right)}$
The result is: $- 40 \sin^{3}{\left(2 x \right)} \sin{\left(8 x \right)} \cos^{4}{\left(8 x \right)} + 6 \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)} \cos^{5}{\left(8 x \right)}$
Now simplify:

$\left(- 40 \sin{\left(2 x \right)} \sin{\left(8 x \right)} + 6 \cos{\left(2 x \right)} \cos{\left(8 x \right)}\right) \sin^{2}{\left(2 x \right)} \cos^{4}{\left(8 x \right)}$

The answer is:

$\left(- 40 \sin{\left(2 x \right)} \sin{\left(8 x \right)} + 6 \cos{\left(2 x \right)} \cos{\left(8 x \right)}\right) \sin^{2}{\left(2 x \right)} \cos^{4}{\left(8 x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        4         3                      5         2              
- 40*cos (8*x)*sin (2*x)*sin(8*x) + 6*cos (8*x)*sin (2*x)*cos(2*x)

$$- 40 \sin^{3}{\left(2 x \right)} \sin{\left(8 x \right)} \cos^{4}{\left(8 x \right)} + 6 \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)} \cos^{5}{\left(8 x \right)}$$

Simplify

The second derivative [src]

     3      /       2      /   2             2     \         2      /     2             2     \                                          \         
4*cos (8*x)*\- 3*cos (8*x)*\sin (2*x) - 2*cos (2*x)/ + 80*sin (2*x)*\- cos (8*x) + 4*sin (8*x)/ - 120*cos(2*x)*cos(8*x)*sin(2*x)*sin(8*x)/*sin(2*x)

$$4 \left(- 120 \sin{\left(2 x \right)} \sin{\left(8 x \right)} \cos{\left(2 x \right)} \cos{\left(8 x \right)} - 3 \left(\sin^{2}{\left(2 x \right)} - 2 \cos^{2}{\left(2 x \right)}\right) \cos^{2}{\left(8 x \right)} + 80 \cdot \left(4 \sin^{2}{\left(8 x \right)} - \cos^{2}{\left(8 x \right)}\right) \sin^{2}{\left(2 x \right)}\right) \sin{\left(2 x \right)} \cos^{3}{\left(8 x \right)}$$

Simplify

The third derivative [src]

     2      /         3      /        2              2     \                 3      /       2             2     \                   2      /   2             2     \                            2      /     2             2     \                  \
8*cos (8*x)*\- 320*sin (2*x)*\- 13*cos (8*x) + 12*sin (8*x)/*sin(8*x) - 3*cos (8*x)*\- 2*cos (2*x) + 7*sin (2*x)/*cos(2*x) + 180*cos (8*x)*\sin (2*x) - 2*cos (2*x)/*sin(2*x)*sin(8*x) + 720*sin (2*x)*\- cos (8*x) + 4*sin (8*x)/*cos(2*x)*cos(8*x)/

$$8 \cdot \left(180 \left(\sin^{2}{\left(2 x \right)} - 2 \cos^{2}{\left(2 x \right)}\right) \sin{\left(2 x \right)} \sin{\left(8 x \right)} \cos^{2}{\left(8 x \right)} - 3 \cdot \left(7 \sin^{2}{\left(2 x \right)} - 2 \cos^{2}{\left(2 x \right)}\right) \cos{\left(2 x \right)} \cos^{3}{\left(8 x \right)} + 720 \cdot \left(4 \sin^{2}{\left(8 x \right)} - \cos^{2}{\left(8 x \right)}\right) \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)} \cos{\left(8 x \right)} - 320 \cdot \left(12 \sin^{2}{\left(8 x \right)} - 13 \cos^{2}{\left(8 x \right)}\right) \sin^{3}{\left(2 x \right)} \sin{\left(8 x \right)}\right) \cos^{2}{\left(8 x \right)}$$

Simplify

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Identical expressions

Derivative of y=sin(2x)^3*cos8x^5

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The solution