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(three *x- four ^ three *sqrt(x)+ two)^ four
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Similar expressions
(3*x-4^3*sqrt(x)-2)^4
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The derivative of the function/ (3*x-4^3*sqrt(x)+2)^4

Derivative of (3x-4^3sqrt(x)+2)^4

The solution

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

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

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

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

Transform

Detail solution

Let $u = - 4^{3} \sqrt{x} + 3 x + 2$ .
Apply the power rule: $u^{4}$ goes to $4 u^{3}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- 4^{3} \sqrt{x} + 3 x + 2\right)$ :
1. Differentiate $- 4^{3} \sqrt{x} + 3 x + 2$ 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: $3$
  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: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
      So, the result is: $\frac{32}{\sqrt{x}}$
    So, the result is: $- \frac{32}{\sqrt{x}}$
  3. The derivative of the constant $2$ is zero.
  The result is: $3 - \frac{32}{\sqrt{x}}$
The result of the chain rule is:

$4 \cdot \left(3 - \frac{32}{\sqrt{x}}\right) \left(- 4^{3} \sqrt{x} + 3 x + 2\right)^{3}$
Now simplify:

$\frac{\left(12 \sqrt{x} - 128\right) \left(- 64 \sqrt{x} + 3 x + 2\right)^{3}}{\sqrt{x}}$

The answer is:

$\frac{\left(12 \sqrt{x} - 128\right) \left(- 64 \sqrt{x} + 3 x + 2\right)^{3}}{\sqrt{x}}$

The graph

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The first derivative [src]

                    3             
/       3   ___    \  /      128 \
\3*x - 4 *\/ x  + 2/ *|12 - -----|
                      |       ___|
                      \     \/ x /

$$\left(12 - \frac{128}{\sqrt{x}}\right) \left(- 4^{3} \sqrt{x} + 3 x + 2\right)^{3}$$

Simplify

The second derivative [src]

                      2 /             2      /         ___      \\
  /         ___      \  |  /      32 \    16*\2 - 64*\/ x  + 3*x/|
4*\2 - 64*\/ x  + 3*x/ *|3*|3 - -----|  + -----------------------|
                        |  |      ___|               3/2         |
                        \  \    \/ x /              x            /

$$4 \cdot \left(3 \left(3 - \frac{32}{\sqrt{x}}\right)^{2} + \frac{16 \left(- 64 \sqrt{x} + 3 x + 2\right)}{x^{\frac{3}{2}}}\right) \left(- 64 \sqrt{x} + 3 x + 2\right)^{2}$$

Simplify

The third derivative [src]

                        /                                            /      32 \ /         ___      \\
                        |                                     2   24*|3 - -----|*\2 - 64*\/ x  + 3*x/|
                        |           3     /         ___      \       |      ___|                     |
   /         ___      \ |/      32 \    4*\2 - 64*\/ x  + 3*x/       \    \/ x /                     |
24*\2 - 64*\/ x  + 3*x/*||3 - -----|  - ----------------------- + -----------------------------------|
                        ||      ___|               5/2                             3/2               |
                        \\    \/ x /              x                               x                  /

$$24 \left(- 64 \sqrt{x} + 3 x + 2\right) \left(\left(3 - \frac{32}{\sqrt{x}}\right)^{3} + \frac{24 \cdot \left(3 - \frac{32}{\sqrt{x}}\right) \left(- 64 \sqrt{x} + 3 x + 2\right)}{x^{\frac{3}{2}}} - \frac{4 \left(- 64 \sqrt{x} + 3 x + 2\right)^{2}}{x^{\frac{5}{2}}}\right)$$

Simplify

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Derivative of (3x-4^3sqrt(x)+2)^4

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The solution

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Derivative of (3*x-4^3*sqrt(x)+2)^4

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The solution

Derivative of (3x-4^3sqrt(x)+2)^4