WordGPT

Derivative of ln⁡(x)x2\frac{\ln\left(x\right)}{x^{2}}

The derivative of ln⁡(x)x2\frac{\ln\left(x\right)}{x^{2}} is x2x−ln⁡(x)2x(x2)2\frac{\frac{x^{2}}{x} - \ln\left(x\right) 2 x}{\left(x^{2}\right)^{2}}.

f(x) =
ddx[ln⁡(x)x2]=x2x−ln⁡(x)2x(x2)2\frac{d}{dx}\left[\frac{\ln\left(x\right)}{x^{2}}\right] = \frac{\frac{x^{2}}{x} - \ln\left(x\right) 2 x}{\left(x^{2}\right)^{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) = ln(x) / x^2 and its derivative-3-2-1123204060slope 1f(x)f'(x)tangent at x = 1
Graph of f(x) = ln(x) / x^2 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)=ln⁡(x)x2f(x) = \frac{\ln\left(x\right)}{x^{2}} with respect to x. The rule that applies is the quotient rule.

    f(x)=ln⁡(x)x2f(x) = \frac{\ln\left(x\right)}{x^{2}}
  2. 2

    Quotient rule: (uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}. Let u=ln⁡(x)u = \ln\left(x\right) and v=x2v = x^{2}.

    u=ln⁡(x),v=x2u = \ln\left(x\right),\quad v = x^{2}
  3. 3

    Differentiate each piece:

    u′=1x,v′=2xu' = \frac{1}{x},\quad v' = 2 x
  4. 4

    Put them in the rule:

    f′(x)=(1x)(x2)−(ln⁡(x))(2x)(x2)2f'(x) = \frac{\left(\frac{1}{x}\right)\left(x^{2}\right) - \left(\ln\left(x\right)\right)\left(2 x\right)}{\left(x^{2}\right)^{2}}
  5. 5

    Simplify:

    f′(x)=x2x−ln⁡(x)2x(x2)2f'(x) = \frac{\frac{x^{2}}{x} - \ln\left(x\right) 2 x}{\left(x^{2}\right)^{2}}
  6. 6

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

    f′′(x)=(2xx−x2x2−(2xx+2ln⁡(x)))(x2)2−(x2x−ln⁡(x)2x)2x22x((x2)2)2f''(x) = \frac{\left(\frac{2 x x - x^{2}}{x^{2}} - \left(\frac{2 x}{x} + 2 \ln\left(x\right)\right)\right) \left(x^{2}\right)^{2} - \left(\frac{x^{2}}{x} - \ln\left(x\right) 2 x\right) 2 x^{2} 2 x}{\left(\left(x^{2}\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\ln\left(x\right)}{x^{2}}\right] = \frac{\frac{x^{2}}{x} - \ln\left(x\right) 2 x}{\left(x^{2}\right)^{2}} \]
\[ f(x) = \frac{\ln\left(x\right)}{x^{2}} \]
\[ u = \ln\left(x\right),\quad v = x^{2} \]
\[ u' = \frac{1}{x},\quad v' = 2 x \]
\[ f'(x) = \frac{\left(\frac{1}{x}\right)\left(x^{2}\right) - \left(\ln\left(x\right)\right)\left(2 x\right)}{\left(x^{2}\right)^{2}} \]
\[ f'(x) = \frac{\frac{x^{2}}{x} - \ln\left(x\right) 2 x}{\left(x^{2}\right)^{2}} \]
\[ f''(x) = \frac{\left(\frac{2 x x - x^{2}}{x^{2}} - \left(\frac{2 x}{x} + 2 \ln\left(x\right)\right)\right) \left(x^{2}\right)^{2} - \left(\frac{x^{2}}{x} - \ln\left(x\right) 2 x\right) 2 x^{2} 2 x}{\left(\left(x^{2}\right)^{2}\right)^{2}} \]

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

xf(x)f'(x)
0.5-2.772588719.090355
101
20.1732868-0.048286795
30.12206803-0.044341651

Questions

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

The derivative of ln⁡(x)x2\frac{\ln\left(x\right)}{x^{2}} is x2x−ln⁡(x)2x(x2)2\frac{\frac{x^{2}}{x} - \ln\left(x\right) 2 x}{\left(x^{2}\right)^{2}}.

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

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

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

The quotient rule.

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

1.

Related derivatives

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