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Identical expressions
sin^ five (x)+cos^ five (x)
sinus of to the power of 5(x) plus co sinus of e of to the power of 5(x)
sinus of to the power of five (x) plus co sinus of e of to the power of five (x)
sin5(x)+cos5(x)
sin5x+cos5x
sin⁵(x)+cos⁵(x)
sin^5x+cos^5x
Similar expressions
sin^5(x)-cos^5(x)

The derivative of the function/ sin^5(x)+cos^5(x)

Derivative of sin^5(x)+cos^5(x)

The solution

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   5         5   
sin (x) + cos (x)

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

d /   5         5   \
--\sin (x) + cos (x)/
dx

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

Transform

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /     5         5           2       3           3       2   \
5*\- cos (x) - sin (x) + 4*cos (x)*sin (x) + 4*cos (x)*sin (x)/

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

Simplify

The third derivative [src]

  /        3            3            2                   2          \              
5*\- 13*sin (x) + 13*cos (x) - 12*sin (x)*cos(x) + 12*cos (x)*sin(x)/*cos(x)*sin(x)

$$5 \left(- 13 \sin^{3}{\left(x \right)} - 12 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 12 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + 13 \cos^{3}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplify

The graph

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Derivative of sin^5(x)+cos^5(x)

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