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Identical expressions
sqrtx*(3x^ two -4x)
square root of x multiply by (3x squared minus 4x)
square root of x multiply by (3x to the power of two minus 4x)
√x*(3x^2-4x)
sqrtx*(3x2-4x)
sqrtx*3x2-4x
sqrtx*(3x²-4x)
sqrtx*(3x to the power of 2-4x)
sqrtx(3x^2-4x)
sqrtx(3x2-4x)
sqrtx3x2-4x
sqrtx3x^2-4x
Similar expressions
sqrtx*(3x^2+4x)

The derivative of the function/ sqrtx*(3x^2-4x)

Derivative of sqrtx*(3x^2-4x)

The solution

You have entered [src]

  ___ /   2      \
\/ x *\3*x  - 4*x/

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

d /  ___ /   2      \\
--\\/ x *\3*x  - 4*x//
dx

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

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sqrt{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
$g{\left(x \right)} = 3 x^{2} - 4 x$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $3 x^{2} - 4 x$ 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: $6 x$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    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: $4$
    So, the result is: $-4$
  The result is: $6 x - 4$
The result is: $\sqrt{x} \left(6 x - 4\right) + \frac{3 x^{2} - 4 x}{2 \sqrt{x}}$
Now simplify:

$\sqrt{x} \left(\frac{15 x}{2} - 6\right)$

The answer is:

$\sqrt{x} \left(\frac{15 x}{2} - 6\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                      2      
  ___              3*x  - 4*x
\/ x *(-4 + 6*x) + ----------
                        ___  
                    2*\/ x

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

Simplify

The second derivative [src]

    ___   2*(-2 + 3*x)   -4 + 3*x
6*\/ x  + ------------ - --------
               ___           ___ 
             \/ x        4*\/ x

$$6 \sqrt{x} - \frac{3 x - 4}{4 \sqrt{x}} + \frac{2 \cdot \left(3 x - 2\right)}{\sqrt{x}}$$

Simplify

The third derivative [src]

  /    -2 + 3*x   -4 + 3*x\
3*|3 - -------- + --------|
  \      2*x        8*x   /
---------------------------
             ___           
           \/ x

$$\frac{3 \cdot \left(3 + \frac{3 x - 4}{8 x} - \frac{3 x - 2}{2 x}\right)}{\sqrt{x}}$$

Simplify

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Derivative of sqrtx*(3x^2-4x)

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The solution