Derivative of $x^{4} \arctan\left(x\right)$

Q: What is the derivative of x^4 * atan(x)?

The derivative of x^4 ≤ft(x) is 4 x^3 ≤ft(x) + x^4 11 + x^2.

Q: What is the second derivative of x^4 * atan(x)?

12 * x^2 * atan(x) + 4 * x^3 * 1 / (1 + x^2) + 4 * x^3 * 1 / (1 + x^2) + x^4 * -(2x) / (1 + x^2)^2.

The derivative of $x^{4} \arctan\left(x\right)$ is $4 x^{3} \arctan\left(x\right) + x^{4} \frac{1}{1 + x^{2}}$ .

f(x) =

\frac{d}{dx}\left[x^{4} \arctan\left(x\right)\right] = 4 x^{3} \arctan\left(x\right) + x^{4} \frac{1}{1 + x^{2}}

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

\frac{d}{dx}\left[x^{4} \arctan\left(x\right)\right] = 4 x^{3} \arctan\left(x\right) + x^{4} \frac{1}{1 + x^{2}}

Explain it step by step in chat Make a short video of it

Graph of f(x) = x^4 * atan(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 3.6416.

How to work it out

1
We differentiate $f(x) = x^{4} \arctan\left(x\right)$ with respect to x. The rule that applies is the product rule.
$f(x) = x^{4} \arctan\left(x\right)$
2
Product rule: $(uv)' = u'v + uv'$ . Let $u = x^{4}$ and $v = \arctan\left(x\right)$ .
$u = x^{4},\quad v = \arctan\left(x\right)$
3
Differentiate each piece:
$u' = 4 x^{3},\quad v' = \frac{1}{1 + x^{2}}$
4
Put them in the rule:
$f'(x) = \left(4 x^{3}\right)\left(\arctan\left(x\right)\right) + \left(x^{4}\right)\left(\frac{1}{1 + x^{2}}\right)$
5
Simplify:
$f'(x) = 4 x^{3} \arctan\left(x\right) + x^{4} \frac{1}{1 + x^{2}}$
6
Differentiating once more gives the second derivative, which tells you about curvature:
$f''(x) = 12 x^{2} \arctan\left(x\right) + 4 x^{3} \frac{1}{1 + x^{2}} + 4 x^{3} \frac{1}{1 + x^{2}} + x^{4} \frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}}$

LaTeX source of the formulas on this page

\[ \frac{d}{dx}\left[x^{4} \arctan\left(x\right)\right] = 4 x^{3} \arctan\left(x\right) + x^{4} \frac{1}{1 + x^{2}} \]
\[ f(x) = x^{4} \arctan\left(x\right) \]
\[ u = x^{4},\quad v = \arctan\left(x\right) \]
\[ u' = 4 x^{3},\quad v' = \frac{1}{1 + x^{2}} \]
\[ f'(x) = \left(4 x^{3}\right)\left(\arctan\left(x\right)\right) + \left(x^{4}\right)\left(\frac{1}{1 + x^{2}}\right) \]
\[ f'(x) = 4 x^{3} \arctan\left(x\right) + x^{4} \frac{1}{1 + x^{2}} \]
\[ f''(x) = 12 x^{2} \arctan\left(x\right) + 4 x^{3} \frac{1}{1 + x^{2}} + 4 x^{3} \frac{1}{1 + x^{2}} + x^{4} \frac{-\left(2 x\right)}{\left(1 + x^{2}\right)^{2}} \]

Values of f and f' for $x^{4}$ * atan(x)

x	f(x)	f'(x)
-2	-17.714379	38.628759
-1	-0.78539816	3.6415927
0.5	0.028977976	0.2818238
1	0.78539816	3.6415927
2	17.714379	38.628759
3	101.17271	142.99694

Questions

What is the derivative of

x^{4}

* atan(x)?

The derivative of $x^{4} \arctan\left(x\right)$ is $4 x^{3} \arctan\left(x\right) + x^{4} \frac{1}{1 + x^{2}}$ .

What is the second derivative of

x^{4}

* atan(x)?

12 * $x^{2}$ * atan(x) + 4 * $x^{3}$ * 1 / (1 + $x^{2}$ ) + 4 * $x^{3}$ * 1 / (1 + $x^{2}$ ) + $x^{4}$ * -(2x) / (1 + $x^{2}$ )^2.

Which rule do you use for

x^{4}

* atan(x)?

The product rule.

What is the slope of

x^{4}

* atan(x) at x = 1?

3.6415927.

Derivative of x4arctan⁡(x)x^{4} \arctan\left(x\right)x4arctan(x)

How to work it out

Values of f and f' for x4x^{4}x4 * atan(x)

Questions

Related derivatives

Derivative of $x^{4} \arctan\left(x\right)$

Values of f and f' for $x^{4}$ * atan(x)