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Derivative of 1xex\frac{1}{x} e^{x}

The derivative of 1xex\frac{1}{x} e^{x} is āˆ’1x2ex+exx\frac{-1}{x^{2}} e^{x} + \frac{e^{x}}{x}.

f(x) =
ddx[1xex]=āˆ’1x2ex+exx\frac{d}{dx}\left[\frac{1}{x} e^{x}\right] = \frac{-1}{x^{2}} e^{x} + \frac{e^{x}}{x}

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Graph of f(x) = 1 / x * exp(x) and its derivative-3-2-1123-15-10-55slope 0f(x)f'(x)tangent at x = 1
Graph of f(x) = 1 / x * exp(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 0.

How to work it out

  1. 1

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

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

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

    u=1x,v=exu = \frac{1}{x},\quad v = e^{x}
  3. 3

    Differentiate each piece:

    uā€²=āˆ’1x2,vā€²=exu' = \frac{-1}{x^{2}},\quad v' = e^{x}
  4. 4

    Put them in the rule:

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

    Simplify:

    fā€²(x)=āˆ’1x2ex+exxf'(x) = \frac{-1}{x^{2}} e^{x} + \frac{e^{x}}{x}
  6. 6

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

    fā€²ā€²(x)=āˆ’((āˆ’2)x)(x2)2ex+āˆ’1x2ex+exxāˆ’exx2f''(x) = \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}} e^{x} + \frac{-1}{x^{2}} e^{x} + \frac{e^{x} x - e^{x}}{x^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{1}{x} e^{x}\right] = \frac{-1}{x^{2}} e^{x} + \frac{e^{x}}{x} \]
\[ f(x) = \frac{1}{x} e^{x} \]
\[ u = \frac{1}{x},\quad v = e^{x} \]
\[ u' = \frac{-1}{x^{2}},\quad v' = e^{x} \]
\[ f'(x) = \left(\frac{-1}{x^{2}}\right)\left(e^{x}\right) + \left(\frac{1}{x}\right)\left(e^{x}\right) \]
\[ f'(x) = \frac{-1}{x^{2}} e^{x} + \frac{e^{x}}{x} \]
\[ f''(x) = \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}} e^{x} + \frac{-1}{x^{2}} e^{x} + \frac{e^{x} x - e^{x}}{x^{2}} \]

Values of f and f' for 1 / x * exp(x)

xf(x)f'(x)
-2-0.067667642-0.10150146
-1-0.36787944-0.73575888
0.53.2974425-3.2974425
12.71828180
23.6945281.847264
36.6951794.4634526

Questions

What is the derivative of 1 / x * exp(x)?

The derivative of 1xex\frac{1}{x} e^{x} is āˆ’1x2ex+exx\frac{-1}{x^{2}} e^{x} + \frac{e^{x}}{x}.

What is the second derivative of 1 / x * exp(x)?

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

Which rule do you use for 1 / x * exp(x)?

The product rule.

What is the slope of 1 / x * exp(x) at x = 1?

0.

Related derivatives

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