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y=x(sqrtx^3)(sqrtx)

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Derivative of:
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Derivative of 4*e^(2*x)
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Derivative of 1^3*sqrt(x)
Identical expressions
y=x(sqrtx^ three)(sqrtx)
y equally x( square root of x cubed )( square root of x)
y equally x( square root of x to the power of three)( square root of x)
y=x(√x^3)(√x)
y=x(sqrtx3)(sqrtx)
y=xsqrtx3sqrtx
y=x(sqrtx³)(sqrtx)
y=x(sqrtx to the power of 3)(sqrtx)
y=xsqrtx^3sqrtx

The derivative of the function/ y=x(sqrtx^3)(sqrtx)

Derivative of y=x(sqrtx^3)(sqrtx)

The solution

You have entered [src]

       3      
    ___    ___
x*\/ x  *\/ x

$$\sqrt{x} x \left(\sqrt{x}\right)^{3}$$

  /       3      \
d |    ___    ___|
--\x*\/ x  *\/ x /
dx

$$\frac{d}{d x} \sqrt{x} x \left(\sqrt{x}\right)^{3}$$

Transform

Detail solution

Apply the product rule:

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

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

$3 x^{2}$

The answer is:

$3 x^{2}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2       ___  3/2
3*x    3*\/ x *x   
---- + ------------
 2          2

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

Simplify

The second derivative [src]

6*x

$$6 x$$

Simplify

The third derivative [src]

$$6$$

Simplify

The graph

Derivative of y=x(sqrtx^3)(sqrtx)