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Derivative of:
Derivative of (x^2-1)/(x*3+1)
Derivative of x^2*sin(x)
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Identical expressions
sqrt(two x+(3x^ four)- five)+ four /(x-2)^ five
square root of (2x plus (3x to the power of 4) minus 5) plus 4 divide by (x minus 2) to the power of 5
square root of (two x plus (3x to the power of four) minus five) plus four divide by (x minus 2) to the power of five
√(2x+(3x^4)-5)+4/(x-2)^5
sqrt(2x+(3x4)-5)+4/(x-2)5
sqrt2x+3x4-5+4/x-25
sqrt(2x+(3x⁴)-5)+4/(x-2)⁵
sqrt2x+3x^4-5+4/x-2^5
sqrt(2x+(3x^4)-5)+4 divide by (x-2)^5
Similar expressions
sqrt(2x+(3x^4)+5)+4/(x-2)^5
sqrt(2x-(3x^4)-5)+4/(x-2)^5
sqrt(2x+(3x^4)-5)+4/(x+2)^5
sqrt(2x+(3x^4)-5)-4/(x-2)^5

The derivative of the function/ sqrt(2x+(3x^4)-5)+4/(x-2)^5

Derivative of sqrt(2x+(3x^4)-5)+4/(x-2)^5

The solution

You have entered [src]

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

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

  /   ________________           \
d |  /          4           4    |
--|\/  2*x + 3*x  - 5  + --------|
dx|                             5|
  \                      (x - 2) /

$$\frac{d}{d x} \left(\sqrt{3 x^{4} + 2 x - 5} + \frac{4}{\left(x - 2\right)^{5}}\right)$$

Transform

Detail solution

Differentiate $\sqrt{3 x^{4} + 2 x - 5} + \frac{4}{\left(x - 2\right)^{5}}$ term by term:
1. Let $u = 3 x^{4} + 2 x - 5$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x^{4} + 2 x - 5\right)$ :
  1. Differentiate $3 x^{4} + 2 x - 5$ term by term:
    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: $2$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
      So, the result is: $12 x^{3}$
    3. The derivative of the constant $\left(-1\right) 5$ is zero.
    The result is: $12 x^{3} + 2$
  The result of the chain rule is:
  
  $\frac{12 x^{3} + 2}{2 \sqrt{3 x^{4} + 2 x - 5}}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \left(x - 2\right)^{5}$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 2\right)^{5}$ :
    1. Let $u = x - 2$ .
    2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 2\right)$ :
      1. Differentiate $x - 2$ term by term:
        
        Apply the power rule: $x$ goes to $1$
        
        The derivative of the constant $\left(-1\right) 2$ is zero.
        
        The result is: $1$
      The result of the chain rule is:
      
      $5 \left(x - 2\right)^{4}$
    The result of the chain rule is:
    
    $- \frac{5}{\left(x - 2\right)^{6}}$
  So, the result is: $- \frac{20}{\left(x - 2\right)^{6}}$
The result is: $\frac{12 x^{3} + 2}{2 \sqrt{3 x^{4} + 2 x - 5}} - \frac{20}{\left(x - 2\right)^{6}}$
Now simplify:

$\frac{6 x^{3}}{\sqrt{3 x^{4} + 2 x - 5}} + \frac{1}{\sqrt{3 x^{4} + 2 x - 5}} - \frac{20}{\left(x - 2\right)^{6}}$

The answer is:

$\frac{6 x^{3}}{\sqrt{3 x^{4} + 2 x - 5}} + \frac{1}{\sqrt{3 x^{4} + 2 x - 5}} - \frac{20}{\left(x - 2\right)^{6}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                          3     
     20            1 + 6*x      
- -------- + -------------------
         6      ________________
  (x - 2)      /          4     
             \/  2*x + 3*x  - 5

$$\frac{6 x^{3} + 1}{\sqrt{3 x^{4} + 2 x - 5}} - \frac{20}{\left(x - 2\right)^{6}}$$

Simplify

The second derivative [src]

                          2                            
                /       3\                    2        
   120          \1 + 6*x /                18*x         
--------- - -------------------- + --------------------
        7                    3/2      _________________
(-2 + x)    /              4\        /               4 
            \-5 + 2*x + 3*x /      \/  -5 + 2*x + 3*x

$$\frac{18 x^{2}}{\sqrt{3 x^{4} + 2 x - 5}} - \frac{\left(6 x^{3} + 1\right)^{2}}{\left(3 x^{4} + 2 x - 5\right)^{\frac{3}{2}}} + \frac{120}{\left(x - 2\right)^{7}}$$

Simplify

The third derivative [src]

  /                            3                                                   \
  |                  /       3\                                      2 /       3\  |
  |     280          \1 + 6*x /                 12*x             18*x *\1 + 6*x /  |
3*|- --------- + -------------------- + -------------------- - --------------------|
  |          8                    5/2      _________________                    3/2|
  |  (-2 + x)    /              4\        /               4    /              4\   |
  \              \-5 + 2*x + 3*x /      \/  -5 + 2*x + 3*x     \-5 + 2*x + 3*x /   /

$$3 \left(- \frac{18 x^{2} \cdot \left(6 x^{3} + 1\right)}{\left(3 x^{4} + 2 x - 5\right)^{\frac{3}{2}}} + \frac{12 x}{\sqrt{3 x^{4} + 2 x - 5}} + \frac{\left(6 x^{3} + 1\right)^{3}}{\left(3 x^{4} + 2 x - 5\right)^{\frac{5}{2}}} - \frac{280}{\left(x - 2\right)^{8}}\right)$$

Simplify

The graph

Derivative of sqrt(2x+(3x^4)-5)+4/(x-2)^5

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Identical expressions

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Derivative of sqrt(2x+(3x^4)-5)+4/(x-2)^5

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The solution