Derivative of $x^{3} \frac{1}{x}$

Q: What is the derivative of x^3 * 1 / x?

The derivative of x^3 (1)/(x) is 3 x^2 (1)/(x) + x^3 -1x^2.

Q: What is the second derivative of x^3 * 1 / x?

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

The derivative of $x^{3} \frac{1}{x}$ is $3 x^{2} \frac{1}{x} + x^{3} \frac{-1}{x^{2}}$ .

f(x) =

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

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

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

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Graph of f(x) = x^3 * 1 / x and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 2.

How to work it out

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

LaTeX source of the formulas on this page

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

Values of f and f' for $x^{3}$ * 1 / x

x	f(x)	f'(x)
-2	4	-4
-1	1	-2
0.5	0.25	1
1	1	2
2	4	4
3	9	6

Questions

What is the derivative of

x^{3}

* 1 / x?

The derivative of $x^{3} \frac{1}{x}$ is $3 x^{2} \frac{1}{x} + x^{3} \frac{-1}{x^{2}}$ .

What is the second derivative of

x^{3}

* 1 / x?

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

Which rule do you use for

x^{3}

* 1 / x?

The product rule.

What is the slope of

x^{3}

* 1 / x at x = 1?

Derivative of x31xx^{3} \frac{1}{x}x3x1​

How to work it out

Values of f and f' for x3x^{3}x3 * 1 / x

Questions

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Derivative of $x^{3} \frac{1}{x}$

Values of f and f' for $x^{3}$ * 1 / x