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Identical expressions
y=(√x^ three +7x)^ three
y equally (√x cubed plus 7x) cubed
y equally (√x to the power of three plus 7x) to the power of three
y=(√x3+7x)3
y=√x3+7x3
y=(√x³+7x)³
y=(√x to the power of 3+7x) to the power of 3
y=√x^3+7x^3
Similar expressions
y=(√x^3-7x)^3

The derivative of the function/ y=(√x^3+7x)^3

Derivative of y=(√x^3+7x)^3

The solution

You have entered [src]

              3
/     3      \ 
|  ___       | 
\\/ x   + 7*x/

$$\left(\left(\sqrt{x}\right)^{3} + 7 x\right)^{3}$$

Transform

Detail solution

Let $u = \left(\sqrt{x}\right)^{3} + 7 x$ .
Apply the power rule: $u^{3}$ goes to $3 u^{2}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(\sqrt{x}\right)^{3} + 7 x\right)$ :
1. Differentiate $\left(\sqrt{x}\right)^{3} + 7 x$ term by term:
  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:
    
    $\frac{3 \sqrt{x}}{2}$
  4. 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: $7$
  The result is: $\frac{3 \sqrt{x}}{2} + 7$
The result of the chain rule is:

$3 \left(\frac{3 \sqrt{x}}{2} + 7\right) \left(\left(\sqrt{x}\right)^{3} + 7 x\right)^{2}$
Now simplify:

$\left(\frac{9 \sqrt{x}}{2} + 21\right) \left(x^{\frac{3}{2}} + 7 x\right)^{2}$

The answer is:

$\left(\frac{9 \sqrt{x}}{2} + 21\right) \left(x^{\frac{3}{2}} + 7 x\right)^{2}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

              2              
/     3      \  /        3/2\
|  ___       |  |     9*x   |
\\/ x   + 7*x/ *|21 + ------|
                \      2*x  /

$$\left(\frac{9 x^{\frac{3}{2}}}{2 x} + 21\right) \left(\left(\sqrt{x}\right)^{3} + 7 x\right)^{2}$$

Simplify

The second derivative [src]

  /                2     / 3/2      \\ / 3/2      \
  |  /         ___\    3*\x    + 7*x/| |x      7*x|
3*|2*\14 + 3*\/ x /  + --------------|*|---- + ---|
  |                          ___     | \ 4      4 /
  \                        \/ x      /

$$3 \left(\frac{x^{\frac{3}{2}}}{4} + \frac{7 x}{4}\right) \left(2 \left(3 \sqrt{x} + 14\right)^{2} + \frac{3 \left(x^{\frac{3}{2}} + 7 x\right)}{\sqrt{x}}\right)$$

Simplify

The third derivative [src]

  /                                  2                                 \
  |                3     / 3/2      \       /         ___\ / 3/2      \|
  |  /         ___\    3*\x    + 7*x/    18*\14 + 3*\/ x /*\x    + 7*x/|
3*|2*\14 + 3*\/ x /  - --------------- + ------------------------------|
  |                           3/2                      ___             |
  \                          x                       \/ x              /
------------------------------------------------------------------------
                                   8

$$\frac{3 \left(2 \left(3 \sqrt{x} + 14\right)^{3} + \frac{18 \left(3 \sqrt{x} + 14\right) \left(x^{\frac{3}{2}} + 7 x\right)}{\sqrt{x}} - \frac{3 \left(x^{\frac{3}{2}} + 7 x\right)^{2}}{x^{\frac{3}{2}}}\right)}{8}$$

Simplify

The graph

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Derivative of y=(√x^3+7x)^3

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The solution