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Identical expressions
sqrt(14x^ two +16x+ twenty-three)
square root of (14x squared plus 16x plus 23)
square root of (14x to the power of two plus 16x plus twenty minus three)
√(14x^2+16x+23)
sqrt(14x2+16x+23)
sqrt14x2+16x+23
sqrt(14x²+16x+23)
sqrt(14x to the power of 2+16x+23)
sqrt14x^2+16x+23
Similar expressions
sqrt(14x^2-16x+23)
sqrt(14x^2+16x-23)

The derivative of the function/ sqrt(14x^2+16x+23)

Derivative of sqrt(14x^2+16x+23)

The solution

You have entered [src]

   ___________________
  /     2             
\/  14*x  + 16*x + 23

$$\sqrt{14 x^{2} + 16 x + 23}$$

  /   ___________________\
d |  /     2             |
--\\/  14*x  + 16*x + 23 /
dx

$$\frac{d}{d x} \sqrt{14 x^{2} + 16 x + 23}$$

Transform

Detail solution

Let $u = 14 x^{2} + 16 x + 23$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(14 x^{2} + 16 x + 23\right)$ :
1. Differentiate $14 x^{2} + 16 x + 23$ 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: $x^{2}$ goes to $2 x$
    So, the result is: $28 x$
  2. 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: $16$
  3. The derivative of the constant $23$ is zero.
  The result is: $28 x + 16$
The result of the chain rule is:

$\frac{28 x + 16}{2 \sqrt{14 x^{2} + 16 x + 23}}$
Now simplify:

$\frac{2 \cdot \left(7 x + 4\right)}{\sqrt{14 x^{2} + 16 x + 23}}$

The answer is:

$\frac{2 \cdot \left(7 x + 4\right)}{\sqrt{14 x^{2} + 16 x + 23}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       8 + 14*x       
----------------------
   ___________________
  /     2             
\/  14*x  + 16*x + 23

$$\frac{14 x + 8}{\sqrt{14 x^{2} + 16 x + 23}}$$

Simplify

The second derivative [src]

  /                  2  \
  |       2*(4 + 7*x)   |
2*|7 - -----------------|
  |             2       |
  \    23 + 14*x  + 16*x/
-------------------------
     ___________________ 
    /          2         
  \/  23 + 14*x  + 16*x

$$\frac{2 \left(- \frac{2 \left(7 x + 4\right)^{2}}{14 x^{2} + 16 x + 23} + 7\right)}{\sqrt{14 x^{2} + 16 x + 23}}$$

Simplify

The third derivative [src]

   /                   2  \          
   |        2*(4 + 7*x)   |          
12*|-7 + -----------------|*(4 + 7*x)
   |              2       |          
   \     23 + 14*x  + 16*x/          
-------------------------------------
                           3/2       
        /         2       \          
        \23 + 14*x  + 16*x/

$$\frac{12 \cdot \left(7 x + 4\right) \left(\frac{2 \left(7 x + 4\right)^{2}}{14 x^{2} + 16 x + 23} - 7\right)}{\left(14 x^{2} + 16 x + 23\right)^{\frac{3}{2}}}$$

Simplify

The graph

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Derivative of sqrt(14x^2+16x+23)

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The solution