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sqrt(x)*(x^4+2)

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Derivative of:
Derivative of f(x)=x^3
Derivative of 6/x^4
Derivative of log(t-1)
Derivative of (5x^2+2)×(x^2-4)^3
Identical expressions
sqrt(x)*(x^ four + two)
square root of (x) multiply by (x to the power of 4 plus 2)
square root of (x) multiply by (x to the power of four plus two)
√(x)*(x^4+2)
sqrt(x)*(x4+2)
sqrtx*x4+2
sqrt(x)*(x⁴+2)
sqrt(x)(x^4+2)
sqrt(x)(x4+2)
sqrtxx4+2
sqrtxx^4+2
Similar expressions
sqrt(x)*(x^4-2)

The derivative of the function/ sqrt(x)*(x^4+2)

Derivative of sqrt(x)*(x^4+2)

The solution

You have entered [src]

  ___ / 4    \
\/ x *\x  + 2/

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

sqrt(x)*(x^4 + 2)

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 $\frac{1}{2 \sqrt{x}}$
$g{\left(x \right)} = x^{4} + 2$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{4} + 2$ term by term:
  1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
  2. The derivative of the constant $2$ is zero.
  The result is: $4 x^{3}$
The result is: $4 x^{\frac{7}{2}} + \frac{x^{4} + 2}{2 \sqrt{x}}$
Now simplify:

$\frac{9 x^{4} + 2}{2 \sqrt{x}}$

The answer is:

$\frac{9 x^{4} + 2}{2 \sqrt{x}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

           4    
   7/2    x  + 2
4*x    + -------
             ___
         2*\/ x

$$4 x^{\frac{7}{2}} + \frac{x^{4} + 2}{2 \sqrt{x}}$$

Simplify

The second derivative [src]

               4
    5/2   2 + x 
16*x    - ------
             3/2
          4*x

$$16 x^{\frac{5}{2}} - \frac{x^{4} + 2}{4 x^{\frac{3}{2}}}$$

Simplify

The third derivative [src]

  /               4\
  |    3/2   2 + x |
3*|13*x    + ------|
  |             5/2|
  \          8*x   /

$$3 \left(13 x^{\frac{3}{2}} + \frac{x^{4} + 2}{8 x^{\frac{5}{2}}}\right)$$

Simplify

The graph

Derivative of sqrt(x)*(x^4+2)