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Derivative of -6/x
Derivative of 5/x^3
Derivative of 3*sin(x)^(2)
Derivative of 2^x*x^2
Identical expressions
(four /x^ five)-(nine /x)+ five ^sqrt(x^ two)
(4 divide by x to the power of 5) minus (9 divide by x) plus 5 to the power of square root of (x squared )
(four divide by x to the power of five) minus (nine divide by x) plus five to the power of square root of (x to the power of two)
(4/x^5)-(9/x)+5^√(x^2)
(4/x5)-(9/x)+5sqrt(x2)
4/x5-9/x+5sqrtx2
(4/x⁵)-(9/x)+5^sqrt(x²)
(4/x to the power of 5)-(9/x)+5 to the power of sqrt(x to the power of 2)
4/x^5-9/x+5^sqrtx^2
(4 divide by x^5)-(9 divide by x)+5^sqrt(x^2)
Similar expressions
(4/x^5)+(9/x)+5^sqrt(x^2)
(4/x^5)-(9/x)-5^sqrt(x^2)

The derivative of the function/ (4/x^5)-(9/x)+5^sqrt(x^2)

Derivative of (4/x^5)-(9/x)+5^sqrt(x^2)

The solution

You have entered [src]

             ____
            /  2 
4    9    \/  x  
-- - - + 5       
 5   x           
x

$$5^{\sqrt{x^{2}}} + \left(\frac{4}{x^{5}} - \frac{9}{x}\right)$$

4/x^5 - 9/x + 5^(sqrt(x^2))

Detail solution

Differentiate $5^{\sqrt{x^{2}}} + \left(\frac{4}{x^{5}} - \frac{9}{x}\right)$ term by term:
1. Differentiate $\frac{4}{x^{5}} - \frac{9}{x}$ term by term:
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = x^{5}$ .
    2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{5}$ :
      1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$
      The result of the chain rule is:
      
      $- \frac{5}{x^{6}}$
    So, the result is: $- \frac{20}{x^{6}}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
    So, the result is: $\frac{9}{x^{2}}$
  The result is: $\frac{9}{x^{2}} - \frac{20}{x^{6}}$
2. Let $u = \sqrt{x^{2}}$ .
3. $\frac{d}{d u} 5^{u} = 5^{u} \log{\left(5 \right)}$
4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{x^{2}}$ :
  1. Let $u = x^{2}$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{2}$ :
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    The result of the chain rule is:
    
    $\frac{x}{\left|{x}\right|}$
  The result of the chain rule is:
  
  $\frac{5^{\sqrt{x^{2}}} x \log{\left(5 \right)}}{\left|{x}\right|}$
The result is: $\frac{5^{\sqrt{x^{2}}} x \log{\left(5 \right)}}{\left|{x}\right|} + \frac{9}{x^{2}} - \frac{20}{x^{6}}$
Now simplify:

$\frac{5^{\left|{x}\right|} x \log{\left(5 \right)}}{\left|{x}\right|} + \frac{9}{x^{2}} - \frac{20}{x^{6}}$

The answer is:

$\frac{5^{\left|{x}\right|} x \log{\left(5 \right)}}{\left|{x}\right|} + \frac{9}{x^{2}} - \frac{20}{x^{6}}$

The first derivative [src]

                ____           
               /  2            
             \/  x             
  20   9    5       *|x|*log(5)
- -- + -- + -------------------
   6    2            x         
  x    x

$$\frac{5^{\sqrt{x^{2}}} \log{\left(5 \right)} \left|{x}\right|}{x} + \frac{9}{x^{2}} - \frac{20}{x^{6}}$$

Simplify

The second derivative [src]

                             |x|                   |x|           
  18   120    |x|    2      5   *log(5)*sign(x)   5   *|x|*log(5)
- -- + --- + 5   *log (5) + ------------------- - ---------------
   3     7                           x                    2      
  x     x                                                x

$$5^{\left|{x}\right|} \log{\left(5 \right)}^{2} + \frac{5^{\left|{x}\right|} \log{\left(5 \right)} \operatorname{sign}{\left(x \right)}}{x} - \frac{5^{\left|{x}\right|} \log{\left(5 \right)} \left|{x}\right|}{x^{2}} - \frac{18}{x^{3}} + \frac{120}{x^{7}}$$

Simplify

The third derivative [src]

                                                                                |x|                     |x|               |x|    2               
  840   54    |x|    2       |x|    3             |x|                        2*5   *log(5)*sign(x)   2*5   *|x|*log(5)   5   *log (5)*|x|*sign(x)
- --- + -- - 5   *log (5) + 5   *log (5)*|x| + 2*5   *DiracDelta(x)*log(5) - --------------------- + ----------------- + ------------------------
    7    3                                                                             x                      2                     x            
   x    x                                                                                                    x                                   
-------------------------------------------------------------------------------------------------------------------------------------------------
                                                                        x

$$\frac{5^{\left|{x}\right|} \log{\left(5 \right)}^{3} \left|{x}\right| + 2 \cdot 5^{\left|{x}\right|} \log{\left(5 \right)} \delta\left(x\right) - 5^{\left|{x}\right|} \log{\left(5 \right)}^{2} + \frac{5^{\left|{x}\right|} \log{\left(5 \right)}^{2} \left|{x}\right| \operatorname{sign}{\left(x \right)}}{x} - \frac{2 \cdot 5^{\left|{x}\right|} \log{\left(5 \right)} \operatorname{sign}{\left(x \right)}}{x} + \frac{2 \cdot 5^{\left|{x}\right|} \log{\left(5 \right)} \left|{x}\right|}{x^{2}} + \frac{54}{x^{3}} - \frac{840}{x^{7}}}{x}$$

Simplify

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Derivative of (4/x^5)-(9/x)+5^sqrt(x^2)

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