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Identical expressions
y=root(one +x)sqrt(x+ three)
y equally root(1 plus x) square root of (x plus 3)
y equally root(one plus x) square root of (x plus three)
y=root(1+x)√(x+3)
y=root1+xsqrtx+3
Similar expressions
y=root(1-x)sqrt(x+3)
y=root(1+x)sqrt(x-3)

The derivative of the function/ y=root(1+x)sqrt(x+3)

Derivative of y=root(1+x)sqrt(x+3)

The solution

You have entered [src]

  _______   _______
\/ 1 + x *\/ x + 3

$$\sqrt{x + 1} \sqrt{x + 3}$$

sqrt(1 + x)*sqrt(x + 3)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \sqrt{x + 1}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = x + 1$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)$ :
  1. Differentiate $x + 1$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result of the chain rule is:
  
  $\frac{1}{2 \sqrt{x + 1}}$
$g{\left(x \right)} = \sqrt{x + 3}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x + 3$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 3\right)$ :
  1. Differentiate $x + 3$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of the constant $3$ is zero.
    The result is: $1$
  The result of the chain rule is:
  
  $\frac{1}{2 \sqrt{x + 3}}$
The result is: $\frac{\sqrt{x + 1}}{2 \sqrt{x + 3}} + \frac{\sqrt{x + 3}}{2 \sqrt{x + 1}}$
Now simplify:

$\frac{x + 2}{\sqrt{x + 1} \sqrt{x + 3}}$

The answer is:

$\frac{x + 2}{\sqrt{x + 1} \sqrt{x + 3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   _______       _______ 
 \/ 1 + x      \/ x + 3  
----------- + -----------
    _______       _______
2*\/ x + 3    2*\/ 1 + x

$$\frac{\sqrt{x + 1}}{2 \sqrt{x + 3}} + \frac{\sqrt{x + 3}}{2 \sqrt{x + 1}}$$

Simplify

The second derivative [src]

    _______      _______                       
  \/ 1 + x     \/ 3 + x              2         
- ---------- - ---------- + -------------------
         3/2          3/2     _______   _______
  (3 + x)      (1 + x)      \/ 1 + x *\/ 3 + x 
-----------------------------------------------
                       4

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

Simplify

The third derivative [src]

  /  _______      _______                                               \
  |\/ 1 + x     \/ 3 + x              1                      1          |
3*|---------- + ---------- - -------------------- - --------------------|
  |       5/2          5/2          3/2   _______     _______        3/2|
  \(3 + x)      (1 + x)      (1 + x)   *\/ 3 + x    \/ 1 + x *(3 + x)   /
-------------------------------------------------------------------------
                                    8

$$\frac{3 \left(\frac{\sqrt{x + 1}}{\left(x + 3\right)^{\frac{5}{2}}} - \frac{1}{\sqrt{x + 1} \left(x + 3\right)^{\frac{3}{2}}} - \frac{1}{\left(x + 1\right)^{\frac{3}{2}} \sqrt{x + 3}} + \frac{\sqrt{x + 3}}{\left(x + 1\right)^{\frac{5}{2}}}\right)}{8}$$

Simplify

The graph

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Derivative of y=root(1+x)sqrt(x+3)

The graph:

Piecewise:

The solution