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

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

f(x) =
ddx[sin⁡(x)ex]=cos⁡(x)ex+sin⁡(x)ex\frac{d}{dx}\left[\sin\left(x\right) e^{x}\right] = \cos\left(x\right) e^{x} + \sin\left(x\right) e^{x}

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Graph of f(x) = sin(x) * exp(x) and its derivative-3-2-1123-10-5510slope 3.756f(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) = 3.756.

How to work it out

  1. 1

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

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

    Product rule: (uv)′=u′v+uv′(uv)' = u'v + uv'. 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)f'(x) = \left(\cos\left(x\right)\right)\left(e^{x}\right) + \left(\sin\left(x\right)\right)\left(e^{x}\right)
  5. 5

    Simplify:

    f′(x)=cos⁡(x)ex+sin⁡(x)exf'(x) = \cos\left(x\right) e^{x} + \sin\left(x\right) e^{x}
  6. 6

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

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

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

xf(x)f'(x)
-2-0.12306002-0.17937937
-1-0.30955988-0.11079377
0.50.790439082.2373281
12.28735533.7560492
26.71884973.6439174
32.8344711-17.05006

Questions

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

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

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).

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

The product rule.

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

3.7560492.

Related derivatives

    Derivative of sin(x) * exp(x): cos(x) * exp(x) + sin(x) * exp(x) (with steps) | WordGPT