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sec(x)+ one / five *x^ five
sec(x) plus 1 divide by 5 multiply by x to the power of 5
sec(x) plus one divide by five multiply by x to the power of five
sec(x)+1/5*x5
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sec(x)+1 divide by 5*x^5
Similar expressions
sec(x)-1/5*x^5

The derivative of the function/ sec(x)+1/5*x^5

Derivative of sec(x)+1/5*x^5

The solution

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          5
         x 
sec(x) + --
         5

$$\frac{x^{5}}{5} + \sec{\left(x \right)}$$

  /          5\
d |         x |
--|sec(x) + --|
dx\         5 /

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

Transform

Detail solution

Differentiate $\frac{x^{5}}{5} + \sec{\left(x \right)}$ term by term:
1. Rewrite the function to be differentiated:
  
  $\sec{\left(x \right)} = \frac{1}{\cos{\left(x \right)}}$
2. Let $u = \cos{\left(x \right)}$ .
3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
4. 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:
  
  $\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
5. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$
  So, the result is: $x^{4}$
The result is: $x^{4} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$

The answer is:

$x^{4} + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 4                
x  + sec(x)*tan(x)

$$x^{4} + \tan{\left(x \right)} \sec{\left(x \right)}$$

Simplify

The second derivative [src]

   3      2             /       2   \       
4*x  + tan (x)*sec(x) + \1 + tan (x)/*sec(x)

$$4 x^{3} + \tan^{2}{\left(x \right)} \sec{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)}$$

Simplify

The third derivative [src]

    2      3               /       2   \              
12*x  + tan (x)*sec(x) + 5*\1 + tan (x)/*sec(x)*tan(x)

$$\tan^{3}{\left(x \right)} \sec{\left(x \right)} + 5 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \sec{\left(x \right)} + 12 x^{2}$$

Simplify

The graph

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Derivative of sec(x)+1/5*x^5

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The solution