WordGPT

Derivative of x21x\frac{x^{2}}{\frac{1}{x}}

The derivative of x21x\frac{x^{2}}{\frac{1}{x}} is 2x1xāˆ’x2āˆ’1x21x2\frac{2 x \frac{1}{x} - x^{2} \frac{-1}{x^{2}}}{\frac{1}{x}^{2}}.

f(x) =
ddx[x21x]=2x1xāˆ’x2āˆ’1x21x2\frac{d}{dx}\left[\frac{x^{2}}{\frac{1}{x}}\right] = \frac{2 x \frac{1}{x} - x^{2} \frac{-1}{x^{2}}}{\frac{1}{x}^{2}}

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

Explain it step by step in chatMake a short video of it
Graph of f(x) = x^2 / (1 / x) and its derivative-3-2-1123-20-101020slope 3f(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) = 3.

How to work it out

  1. 1

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

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

    Quotient rule: (uv)ā€²=uā€²vāˆ’uvā€²v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}. 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)āˆ’(x2)(āˆ’1x2)(1x)2f'(x) = \frac{\left(2 x\right)\left(\frac{1}{x}\right) - \left(x^{2}\right)\left(\frac{-1}{x^{2}}\right)}{\left(\frac{1}{x}\right)^{2}}
  5. 5

    Simplify:

    fā€²(x)=2x1xāˆ’x2āˆ’1x21x2f'(x) = \frac{2 x \frac{1}{x} - x^{2} \frac{-1}{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’1x2āˆ’(2xāˆ’1x2+x2āˆ’((āˆ’2)x)(x2)2))1x2āˆ’(2x1xāˆ’x2āˆ’1x2)21xāˆ’1x2(1x2)2f''(x) = \frac{\left(2 \frac{1}{x} + 2 x \frac{-1}{x^{2}} - \left(2 x \frac{-1}{x^{2}} + x^{2} \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}\right)\right) \frac{1}{x}^{2} - \left(2 x \frac{1}{x} - x^{2} \frac{-1}{x^{2}}\right) 2 \frac{1}{x} \frac{-1}{x^{2}}}{\left(\frac{1}{x}^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{x^{2}}{\frac{1}{x}}\right] = \frac{2 x \frac{1}{x} - x^{2} \frac{-1}{x^{2}}}{\frac{1}{x}^{2}} \]
\[ f(x) = \frac{x^{2}}{\frac{1}{x}} \]
\[ u = x^{2},\quad v = \frac{1}{x} \]
\[ u' = 2 x,\quad v' = \frac{-1}{x^{2}} \]
\[ f'(x) = \frac{\left(2 x\right)\left(\frac{1}{x}\right) - \left(x^{2}\right)\left(\frac{-1}{x^{2}}\right)}{\left(\frac{1}{x}\right)^{2}} \]
\[ f'(x) = \frac{2 x \frac{1}{x} - x^{2} \frac{-1}{x^{2}}}{\frac{1}{x}^{2}} \]
\[ f''(x) = \frac{\left(2 \frac{1}{x} + 2 x \frac{-1}{x^{2}} - \left(2 x \frac{-1}{x^{2}} + x^{2} \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}\right)\right) \frac{1}{x}^{2} - \left(2 x \frac{1}{x} - x^{2} \frac{-1}{x^{2}}\right) 2 \frac{1}{x} \frac{-1}{x^{2}}}{\left(\frac{1}{x}^{2}\right)^{2}} \]

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

xf(x)f'(x)
-2-812
-1-13
0.50.1250.75
113
2812
32727

Questions

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

The derivative of x21x\frac{x^{2}}{\frac{1}{x}} is 2x1xāˆ’x2āˆ’1x21x2\frac{2 x \frac{1}{x} - x^{2} \frac{-1}{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)) * (1 / x)^2 - (2x * 1 / x - x2x^{2} * -1 / x2x^{2}) * 2 * 1 / x * -1 / x2x^{2}) / ((1 / x)^2)^2.

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

The quotient rule.

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

3.

Related derivatives

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