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Derivative of (x^3+2x)^37
Identical expressions
(x^ three +2x)^ thirty-seven
(x cubed plus 2x) cubed 7
(x to the power of three plus 2x) to the power of thirty minus seven
(x3+2x)37
x3+2x37
(x³+2x)³7
(x to the power of 3+2x) to the power of 37
x^3+2x^37
Similar expressions
(x^3-2x)^37

The derivative of the function/ (x^3+2x)^37

Derivative of (x^3+2x)^37

The solution

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          37
/ 3      \  
\x  + 2*x/

$$\left(x^{3} + 2 x\right)^{37}$$

(x^3 + 2*x)^37

Detail solution

Let $u = x^{3} + 2 x$ .
Apply the power rule: $u^{37}$ goes to $37 u^{36}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{3} + 2 x\right)$ :
1. Differentiate $x^{3} + 2 x$ term by term:
  1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
  2. 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: $2$
  The result is: $3 x^{2} + 2$
The result of the chain rule is:

$37 \left(3 x^{2} + 2\right) \left(x^{3} + 2 x\right)^{36}$
Now simplify:

$x^{36} \left(x^{2} + 2\right)^{36} \left(111 x^{2} + 74\right)$

The answer is:

$x^{36} \left(x^{2} + 2\right)^{36} \left(111 x^{2} + 74\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

          36              
/ 3      \   /          2\
\x  + 2*x/  *\74 + 111*x /

$$\left(111 x^{2} + 74\right) \left(x^{3} + 2 x\right)^{36}$$

Simplify

The second derivative [src]

                35 /            2              \
     35 /     2\   |  /       2\     2 /     2\|
222*x  *\2 + x /  *\6*\2 + 3*x /  + x *\2 + x //

$$222 x^{35} \left(x^{2} + 2\right)^{35} \left(x^{2} \left(x^{2} + 2\right) + 6 \left(3 x^{2} + 2\right)^{2}\right)$$

Simplify

The third derivative [src]

                34 /              3              2                             \
     34 /     2\   |    /       2\     2 /     2\         2 /     2\ /       2\|
222*x  *\2 + x /  *\210*\2 + 3*x /  + x *\2 + x /  + 108*x *\2 + x /*\2 + 3*x //

$$222 x^{34} \left(x^{2} + 2\right)^{34} \left(x^{2} \left(x^{2} + 2\right)^{2} + 108 x^{2} \left(x^{2} + 2\right) \left(3 x^{2} + 2\right) + 210 \left(3 x^{2} + 2\right)^{3}\right)$$

Simplify

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Derivative of (x^3+2x)^37

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The solution