Derivative of $x^{4} \frac{1}{x}$

Q: What is the derivative of x^4 * 1 / x?

The derivative of x^4 (1)/(x) is 4 x^3 (1)/(x) + x^4 -1x^2.

Q: What is the second derivative of x^4 * 1 / x?

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

The derivative of $x^{4} \frac{1}{x}$ is $4 x^{3} \frac{1}{x} + x^{4} \frac{-1}{x^{2}}$ .

f(x) =

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

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

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

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

Graph of f(x) = x^4 * 1 / x and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 3.

How to work it out

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

LaTeX source of the formulas on this page

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

Values of f and f' for $x^{4}$ * 1 / x

x	f(x)	f'(x)
-2	-8	12
-1	-1	3
0.5	0.125	0.75
1	1	3
2	8	12
3	27	27

Questions

What is the derivative of

x^{4}

* 1 / x?

The derivative of $x^{4} \frac{1}{x}$ is $4 x^{3} \frac{1}{x} + x^{4} \frac{-1}{x^{2}}$ .

What is the second derivative of

x^{4}

* 1 / x?

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

Which rule do you use for

x^{4}

* 1 / x?

The product rule.

What is the slope of

x^{4}

* 1 / x at x = 1?

Derivative of x41xx^{4} \frac{1}{x}x4x1​

How to work it out

Values of f and f' for x4x^{4}x4 * 1 / x

Questions

Related derivatives

Derivative of $x^{4} \frac{1}{x}$

Values of f and f' for $x^{4}$ * 1 / x