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Derivative of sin⁡(x)ex\frac{\sin\left(x\right)}{e^{x}}

The derivative of sin⁡(x)ex\frac{\sin\left(x\right)}{e^{x}} is \frac{\cos\left(x\right) e^{x} - \sin\left(x\right) e^{x}}{e^{x}^{2}}.

f(x) =
\frac{d}{dx}\left[\frac{\sin\left(x\right)}{e^{x}}\right] = \frac{\cos\left(x\right) e^{x} - \sin\left(x\right) e^{x}}{e^{x}^{2}}

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Graph of f(x) = sin(x) / exp(x) and its derivative-3-2-1123-8-6-4-2246slope -0.1108f(x)f'(x)tangent at x = 1
Graph of f(x) = sin(x) / exp(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = -0.11079.

How to work it out

  1. 1

    We differentiate f(x)=sin⁡(x)exf(x) = \frac{\sin\left(x\right)}{e^{x}} with respect to x. The rule that applies is the quotient rule.

    f(x)=sin⁡(x)exf(x) = \frac{\sin\left(x\right)}{e^{x}}
  2. 2

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

    u=sin⁡(x),v=exu = \sin\left(x\right),\quad v = e^{x}
  3. 3

    Differentiate each piece:

    u′=cos⁡(x),v′=exu' = \cos\left(x\right),\quad v' = e^{x}
  4. 4

    Put them in the rule:

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

    Simplify:

    f'(x) = \frac{\cos\left(x\right) e^{x} - \sin\left(x\right) e^{x}}{e^{x}^{2}}
  6. 6

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

    f''(x) = \frac{\left(-\left(\sin\left(x\right) e^{x}\right) + \cos\left(x\right) e^{x} - \left(\cos\left(x\right) e^{x} + \sin\left(x\right) e^{x}\right)\right) e^{x}^{2} - \left(\cos\left(x\right) e^{x} - \sin\left(x\right) e^{x}\right) 2 e^{x} e^{x}}{\left(e^{x}^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\sin\left(x\right)}{e^{x}}\right] = \frac{\cos\left(x\right) e^{x} - \sin\left(x\right) e^{x}}{e^{x}^{2}} \]
\[ f(x) = \frac{\sin\left(x\right)}{e^{x}} \]
\[ u = \sin\left(x\right),\quad v = e^{x} \]
\[ u' = \cos\left(x\right),\quad v' = e^{x} \]
\[ f'(x) = \frac{\left(\cos\left(x\right)\right)\left(e^{x}\right) - \left(\sin\left(x\right)\right)\left(e^{x}\right)}{\left(e^{x}\right)^{2}} \]
\[ f'(x) = \frac{\cos\left(x\right) e^{x} - \sin\left(x\right) e^{x}}{e^{x}^{2}} \]
\[ f''(x) = \frac{\left(-\left(\sin\left(x\right) e^{x}\right) + \cos\left(x\right) e^{x} - \left(\cos\left(x\right) e^{x} + \sin\left(x\right) e^{x}\right)\right) e^{x}^{2} - \left(\cos\left(x\right) e^{x} - \sin\left(x\right) e^{x}\right) 2 e^{x} e^{x}}{\left(e^{x}^{2}\right)^{2}} \]

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

xf(x)f'(x)
-2-6.71884973.6439174
-1-2.28735533.7560492
0.50.290786290.24149444
10.30955988-0.11079377
20.12306002-0.17937937
30.0070259515-0.056314776

Questions

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

The derivative of sin⁡(x)ex\frac{\sin\left(x\right)}{e^{x}} is \frac{\cos\left(x\right) e^{x} - \sin\left(x\right) e^{x}}{e^{x}^{2}}.

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

((-(sin(x) * exp(x)) + cos(x) * exp(x) - (cos(x) * exp(x) + sin(x) * exp(x))) * exp(x)^2 - (cos(x) * exp(x) - sin(x) * exp(x)) * 2exp(x) * exp(x)) / (exp(x)^2)^2.

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

The quotient rule.

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

-0.11079377.

Related derivatives

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