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Derivative of x21xx^{2} \frac{1}{x}

The derivative of x21xx^{2} \frac{1}{x} is 2x1x+x2āˆ’1x22 x \frac{1}{x} + x^{2} \frac{-1}{x^{2}}.

f(x) =
ddx[x21x]=2x1x+x2āˆ’1x2\frac{d}{dx}\left[x^{2} \frac{1}{x}\right] = 2 x \frac{1}{x} + x^{2} \frac{-1}{x^{2}}

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Graph of f(x) = x^2 * 1 / x and its derivative-3-2-1123-3-2-1123slope 1f(x)f'(x)tangent at x = 1
Graph of f(x) = x^2 * 1 / x and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 1.

How to work it out

  1. 1

    We differentiate f(x)=x21xf(x) = x^{2} \frac{1}{x} with respect to x. The rule that applies is the product rule.

    f(x)=x21xf(x) = x^{2} \frac{1}{x}
  2. 2

    Product rule: (uv)ā€²=uā€²v+uvā€²(uv)' = u'v + uv'. Let u=x2u = x^{2} and v=1xv = \frac{1}{x}.

    u=x2,v=1xu = x^{2},\quad v = \frac{1}{x}
  3. 3

    Differentiate each piece:

    uā€²=2x,vā€²=āˆ’1x2u' = 2 x,\quad v' = \frac{-1}{x^{2}}
  4. 4

    Put them in the rule:

    fā€²(x)=(2x)(1x)+(x2)(āˆ’1x2)f'(x) = \left(2 x\right)\left(\frac{1}{x}\right) + \left(x^{2}\right)\left(\frac{-1}{x^{2}}\right)
  5. 5

    Simplify:

    fā€²(x)=2x1x+x2āˆ’1x2f'(x) = 2 x \frac{1}{x} + x^{2} \frac{-1}{x^{2}}
  6. 6

    Differentiating once more gives the second derivative, which tells you about curvature:

    fā€²ā€²(x)=21x+2xāˆ’1x2+2xāˆ’1x2+x2āˆ’((āˆ’2)x)(x2)2f''(x) = 2 \frac{1}{x} + 2 x \frac{-1}{x^{2}} + 2 x \frac{-1}{x^{2}} + x^{2} \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[x^{2} \frac{1}{x}\right] = 2 x \frac{1}{x} + x^{2} \frac{-1}{x^{2}} \]
\[ f(x) = x^{2} \frac{1}{x} \]
\[ u = x^{2},\quad v = \frac{1}{x} \]
\[ u' = 2 x,\quad v' = \frac{-1}{x^{2}} \]
\[ f'(x) = \left(2 x\right)\left(\frac{1}{x}\right) + \left(x^{2}\right)\left(\frac{-1}{x^{2}}\right) \]
\[ f'(x) = 2 x \frac{1}{x} + x^{2} \frac{-1}{x^{2}} \]
\[ f''(x) = 2 \frac{1}{x} + 2 x \frac{-1}{x^{2}} + 2 x \frac{-1}{x^{2}} + x^{2} \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}} \]

Values of f and f' for x2x^{2} * 1 / x

xf(x)f'(x)
-2-21
-1-11
0.50.51
111
221
331

Questions

What is the derivative of x2x^{2} * 1 / x?

The derivative of x21xx^{2} \frac{1}{x} is 2x1x+x2āˆ’1x22 x \frac{1}{x} + x^{2} \frac{-1}{x^{2}}.

What is the second derivative of x2x^{2} * 1 / x?

2 * 1 / x + 2x * -1 / x2x^{2} + 2x * -1 / x2x^{2} + x2x^{2} * -(-2x) / (x2x^{2})^2.

Which rule do you use for x2x^{2} * 1 / x?

The product rule.

What is the slope of x2x^{2} * 1 / x at x = 1?

1.

Related derivatives

    Derivative of x^2 * 1 / x: 2x * 1 / x + x^2 * -1 / x^2 (with steps) | WordGPT