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Derivative of:
Derivative of x^4+4*x^3-8*x^2-5
Derivative of x^(3/2)-3*x+1
Derivative of x^2*(x-2)
Derivative of -(x^2+289)/x
Identical expressions
y=n*4x*sqrt(2x+ three)
y equally n multiply by 4x multiply by square root of (2x plus 3)
y equally n multiply by 4x multiply by square root of (2x plus three)
y=n*4x*√(2x+3)
y=n4xsqrt(2x+3)
y=n4xsqrt2x+3
Similar expressions
y=n*4x*sqrt(2x-3)

The derivative of the function/ y=n*4x*sqrt(2x+3)

Derivative of y=n4xsqrt(2x+3)

The solution

You have entered [src]

        _________
n*4*x*\/ 2*x + 3

$$x 4 n \sqrt{2 x + 3}$$

((n*4)*x)*sqrt(2*x + 3)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x 4 n$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
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: $4 n$
$g{\left(x \right)} = \sqrt{2 x + 3}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x + 3$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 3\right)$ :
  1. Differentiate $2 x + 3$ 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 the constant $3$ is zero.
    The result is: $2$
  The result of the chain rule is:
  
  $\frac{1}{\sqrt{2 x + 3}}$
The result is: $\frac{4 n x}{\sqrt{2 x + 3}} + 4 n \sqrt{2 x + 3}$
Now simplify:

$\frac{12 n \left(x + 1\right)}{\sqrt{2 x + 3}}$

The answer is:

$\frac{12 n \left(x + 1\right)}{\sqrt{2 x + 3}}$

The first derivative [src]

      _________      4*n*x   
4*n*\/ 2*x + 3  + -----------
                    _________
                  \/ 2*x + 3

$$\frac{4 n x}{\sqrt{2 x + 3}} + 4 n \sqrt{2 x + 3}$$

Simplify

The second derivative [src]

    /       x   \
4*n*|2 - -------|
    \    3 + 2*x/
-----------------
     _________   
   \/ 3 + 2*x

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

Simplify

The third derivative [src]

     /        x   \
12*n*|-1 + -------|
     \     3 + 2*x/
-------------------
             3/2   
    (3 + 2*x)

$$\frac{12 n \left(\frac{x}{2 x + 3} - 1\right)}{\left(2 x + 3\right)^{\frac{3}{2}}}$$

Simplify