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Derivative of exx2\frac{e^{x}}{x^{2}}

The derivative of exx2\frac{e^{x}}{x^{2}} is exx2−ex2x(x2)2\frac{e^{x} x^{2} - e^{x} 2 x}{\left(x^{2}\right)^{2}}.

f(x) =
ddx[exx2]=exx2−ex2x(x2)2\frac{d}{dx}\left[\frac{e^{x}}{x^{2}}\right] = \frac{e^{x} x^{2} - e^{x} 2 x}{\left(x^{2}\right)^{2}}

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Graph of f(x) = exp(x) / x^2 and its derivative-3-2-1123-20204060slope -2.718f(x)f'(x)tangent at x = 1
Graph of f(x) = exp(x) / x^2 and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -2.7183.

How to work it out

  1. 1

    We differentiate f(x)=exx2f(x) = \frac{e^{x}}{x^{2}} with respect to x. The rule that applies is the quotient rule.

    f(x)=exx2f(x) = \frac{e^{x}}{x^{2}}
  2. 2

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

    u=ex,v=x2u = e^{x},\quad v = x^{2}
  3. 3

    Differentiate each piece:

    u′=ex,v′=2xu' = e^{x},\quad v' = 2 x
  4. 4

    Put them in the rule:

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

    Simplify:

    f′(x)=exx2−ex2x(x2)2f'(x) = \frac{e^{x} x^{2} - e^{x} 2 x}{\left(x^{2}\right)^{2}}
  6. 6

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

    f′′(x)=(exx2+ex2x−(ex2x+2ex))(x2)2−(exx2−ex2x)2x22x((x2)2)2f''(x) = \frac{\left(e^{x} x^{2} + e^{x} 2 x - \left(e^{x} 2 x + 2 e^{x}\right)\right) \left(x^{2}\right)^{2} - \left(e^{x} x^{2} - e^{x} 2 x\right) 2 x^{2} 2 x}{\left(\left(x^{2}\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{e^{x}}{x^{2}}\right] = \frac{e^{x} x^{2} - e^{x} 2 x}{\left(x^{2}\right)^{2}} \]
\[ f(x) = \frac{e^{x}}{x^{2}} \]
\[ u = e^{x},\quad v = x^{2} \]
\[ u' = e^{x},\quad v' = 2 x \]
\[ f'(x) = \frac{\left(e^{x}\right)\left(x^{2}\right) - \left(e^{x}\right)\left(2 x\right)}{\left(x^{2}\right)^{2}} \]
\[ f'(x) = \frac{e^{x} x^{2} - e^{x} 2 x}{\left(x^{2}\right)^{2}} \]
\[ f''(x) = \frac{\left(e^{x} x^{2} + e^{x} 2 x - \left(e^{x} 2 x + 2 e^{x}\right)\right) \left(x^{2}\right)^{2} - \left(e^{x} x^{2} - e^{x} 2 x\right) 2 x^{2} 2 x}{\left(\left(x^{2}\right)^{2}\right)^{2}} \]

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

xf(x)f'(x)
-20.0338338210.067667642
-10.367879441.1036383
0.56.5948851-19.784655
12.7182818-2.7182818
21.8472640
32.23172630.74390877

Questions

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

The derivative of exx2\frac{e^{x}}{x^{2}} is exx2−ex2x(x2)2\frac{e^{x} x^{2} - e^{x} 2 x}{\left(x^{2}\right)^{2}}.

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

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

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

The quotient rule.

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

-2.7182818.

Related derivatives

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