Derivative of $x^{3} \arctan\left(x\right)$

Q: What is the derivative of x^3 * atan(x)?

The derivative of x^3 ≤ft(x) is 3 x^2 ≤ft(x) + x^3 11 + x^2.

Q: What is the second derivative of x^3 * atan(x)?

6x * atan(x) + 3 * x^2 * 1 / (1 + x^2) + 3 * x^2 * 1 / (1 + x^2) + x^3 * -(2x) / (1 + x^2)^2.

The derivative of $x^{3} \arctan\left(x\right)$ is $3 x^{2} \arctan\left(x\right) + x^{3} \frac{1}{1 + x^{2}}$ .

f(x) =

\frac{d}{dx}\left[x^{3} \arctan\left(x\right)\right] = 3 x^{2} \arctan\left(x\right) + x^{3} \frac{1}{1 + x^{2}}

Type any function of x; the derivative is computed symbolically in your browser.

\frac{d}{dx}\left[x^{3} \arctan\left(x\right)\right] = 3 x^{2} \arctan\left(x\right) + x^{3} \frac{1}{1 + x^{2}}

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Graph of f(x) = x^3 * atan(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 2.8562.

How to work it out

1
We differentiate $f(x) = x^{3} \arctan\left(x\right)$ with respect to x. The rule that applies is the product rule.
$f(x) = x^{3} \arctan\left(x\right)$
2
Product rule: $(uv)' = u'v + uv'$ . Let $u = x^{3}$ and $v = \arctan\left(x\right)$ .
$u = x^{3},\quad v = \arctan\left(x\right)$
3
Differentiate each piece:
$u' = 3 x^{2},\quad v' = \frac{1}{1 + x^{2}}$
4
Put them in the rule:
$f'(x) = \left(3 x^{2}\right)\left(\arctan\left(x\right)\right) + \left(x^{3}\right)\left(\frac{1}{1 + x^{2}}\right)$
5
Simplify:
$f'(x) = 3 x^{2} \arctan\left(x\right) + x^{3} \frac{1}{1 + x^{2}}$
6
Differentiating once more gives the second derivative, which tells you about curvature:
$f''(x) = 6 x \arctan\left(x\right) + 3 x^{2} \frac{1}{1 + x^{2}} + 3 x^{2} \frac{1}{1 + x^{2}} + x^{3} \frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}}$

LaTeX source of the formulas on this page

\[ \frac{d}{dx}\left[x^{3} \arctan\left(x\right)\right] = 3 x^{2} \arctan\left(x\right) + x^{3} \frac{1}{1 + x^{2}} \]
\[ f(x) = x^{3} \arctan\left(x\right) \]
\[ u = x^{3},\quad v = \arctan\left(x\right) \]
\[ u' = 3 x^{2},\quad v' = \frac{1}{1 + x^{2}} \]
\[ f'(x) = \left(3 x^{2}\right)\left(\arctan\left(x\right)\right) + \left(x^{3}\right)\left(\frac{1}{1 + x^{2}}\right) \]
\[ f'(x) = 3 x^{2} \arctan\left(x\right) + x^{3} \frac{1}{1 + x^{2}} \]
\[ f''(x) = 6 x \arctan\left(x\right) + 3 x^{2} \frac{1}{1 + x^{2}} + 3 x^{2} \frac{1}{1 + x^{2}} + x^{3} \frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}} \]

Values of f and f' for $x^{3}$ * atan(x)

x	f(x)	f'(x)
-2	8.8571897	-14.885785
-1	0.78539816	-2.8561945
0.5	0.057955951	0.44773571
1	0.78539816	2.8561945
2	8.8571897	14.885785
3	33.724236	36.424236

Questions

What is the derivative of

x^{3}

* atan(x)?

The derivative of $x^{3} \arctan\left(x\right)$ is $3 x^{2} \arctan\left(x\right) + x^{3} \frac{1}{1 + x^{2}}$ .

What is the second derivative of

x^{3}

* atan(x)?

6x * atan(x) + 3 * $x^{2}$ * 1 / (1 + $x^{2}$ ) + 3 * $x^{2}$ * 1 / (1 + $x^{2}$ ) + $x^{3}$ * -(2x) / (1 + $x^{2}$ )^2.

Which rule do you use for

x^{3}

* atan(x)?

The product rule.

What is the slope of

x^{3}

* atan(x) at x = 1?

2.8561945.

Derivative of x3arctan⁡(x)x^{3} \arctan\left(x\right)x3arctan(x)

How to work it out

Values of f and f' for x3x^{3}x3 * atan(x)

Questions

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Derivative of $x^{3} \arctan\left(x\right)$

Values of f and f' for $x^{3}$ * atan(x)