WordGPT

Derivative of ex1x\frac{e^{x}}{\frac{1}{x}}

The derivative of ex1x\frac{e^{x}}{\frac{1}{x}} is ex1xāˆ’exāˆ’1x21x2\frac{e^{x} \frac{1}{x} - e^{x} \frac{-1}{x^{2}}}{\frac{1}{x}^{2}}.

f(x) =
ddx[ex1x]=ex1xāˆ’exāˆ’1x21x2\frac{d}{dx}\left[\frac{e^{x}}{\frac{1}{x}}\right] = \frac{e^{x} \frac{1}{x} - e^{x} \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) = exp(x) / (1 / x) and its derivative-3-2-1123-20-101020304050slope 5.437f(x)f'(x)tangent at x = 1
Graph of f(x) = exp(x) / (1 / x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 5.4366.

How to work it out

  1. 1

    We differentiate f(x)=ex1xf(x) = \frac{e^{x}}{\frac{1}{x}} with respect to x. The rule that applies is the quotient rule.

    f(x)=ex1xf(x) = \frac{e^{x}}{\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=exu = e^{x} and v=1xv = \frac{1}{x}.

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

    Differentiate each piece:

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

    Put them in the rule:

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

    Simplify:

    fā€²(x)=ex1xāˆ’exāˆ’1x21x2f'(x) = \frac{e^{x} \frac{1}{x} - e^{x} \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)=(ex1x+exāˆ’1x2āˆ’(exāˆ’1x2+exāˆ’((āˆ’2)x)(x2)2))1x2āˆ’(ex1xāˆ’exāˆ’1x2)21xāˆ’1x2(1x2)2f''(x) = \frac{\left(e^{x} \frac{1}{x} + e^{x} \frac{-1}{x^{2}} - \left(e^{x} \frac{-1}{x^{2}} + e^{x} \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}\right)\right) \frac{1}{x}^{2} - \left(e^{x} \frac{1}{x} - e^{x} \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
\[ \frac{d}{dx}\left[\frac{e^{x}}{\frac{1}{x}}\right] = \frac{e^{x} \frac{1}{x} - e^{x} \frac{-1}{x^{2}}}{\frac{1}{x}^{2}} \]
\[ f(x) = \frac{e^{x}}{\frac{1}{x}} \]
\[ u = e^{x},\quad v = \frac{1}{x} \]
\[ u' = e^{x},\quad v' = \frac{-1}{x^{2}} \]
\[ f'(x) = \frac{\left(e^{x}\right)\left(\frac{1}{x}\right) - \left(e^{x}\right)\left(\frac{-1}{x^{2}}\right)}{\left(\frac{1}{x}\right)^{2}} \]
\[ f'(x) = \frac{e^{x} \frac{1}{x} - e^{x} \frac{-1}{x^{2}}}{\frac{1}{x}^{2}} \]
\[ f''(x) = \frac{\left(e^{x} \frac{1}{x} + e^{x} \frac{-1}{x^{2}} - \left(e^{x} \frac{-1}{x^{2}} + e^{x} \frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}\right)\right) \frac{1}{x}^{2} - \left(e^{x} \frac{1}{x} - e^{x} \frac{-1}{x^{2}}\right) 2 \frac{1}{x} \frac{-1}{x^{2}}}{\left(\frac{1}{x}^{2}\right)^{2}} \]

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

xf(x)f'(x)
-2-0.27067057-0.13533528
-1-0.367879440
0.50.824360642.4730819
12.71828185.4365637
214.77811222.167168
360.25661180.342148

Questions

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

The derivative of ex1x\frac{e^{x}}{\frac{1}{x}} is ex1xāˆ’exāˆ’1x21x2\frac{e^{x} \frac{1}{x} - e^{x} \frac{-1}{x^{2}}}{\frac{1}{x}^{2}}.

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

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

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

The quotient rule.

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

5.4365637.

Related derivatives

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