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Derivative of sin⁡(x)ln⁡(x)\sin\left(x\right) \ln\left(x\right)

The derivative of sin⁡(x)ln⁡(x)\sin\left(x\right) \ln\left(x\right) is cos⁡(x)ln⁡(x)+sin⁡(x)1x\cos\left(x\right) \ln\left(x\right) + \sin\left(x\right) \frac{1}{x}.

f(x) =
ddx[sin⁡(x)ln⁡(x)]=cos⁡(x)ln⁡(x)+sin⁡(x)1x\frac{d}{dx}\left[\sin\left(x\right) \ln\left(x\right)\right] = \cos\left(x\right) \ln\left(x\right) + \sin\left(x\right) \frac{1}{x}

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Graph of f(x) = sin(x) * ln(x) and its derivative-3-2-1123-3-2-112slope 0.8415f(x)f'(x)tangent at x = 1
Graph of f(x) = sin(x) * ln(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 0.84147.

How to work it out

  1. 1

    We differentiate f(x)=sin⁡(x)ln⁡(x)f(x) = \sin\left(x\right) \ln\left(x\right) with respect to x. The rule that applies is the product rule.

    f(x)=sin⁡(x)ln⁡(x)f(x) = \sin\left(x\right) \ln\left(x\right)
  2. 2

    Product rule: (uv)′=u′v+uv′(uv)' = u'v + uv'. Let u=sin⁡(x)u = \sin\left(x\right) and v=ln⁡(x)v = \ln\left(x\right).

    u=sin⁡(x),v=ln⁡(x)u = \sin\left(x\right),\quad v = \ln\left(x\right)
  3. 3

    Differentiate each piece:

    u′=cos⁡(x),v′=1xu' = \cos\left(x\right),\quad v' = \frac{1}{x}
  4. 4

    Put them in the rule:

    f′(x)=(cos⁡(x))(ln⁡(x))+(sin⁡(x))(1x)f'(x) = \left(\cos\left(x\right)\right)\left(\ln\left(x\right)\right) + \left(\sin\left(x\right)\right)\left(\frac{1}{x}\right)
  5. 5

    Simplify:

    f′(x)=cos⁡(x)ln⁡(x)+sin⁡(x)1xf'(x) = \cos\left(x\right) \ln\left(x\right) + \sin\left(x\right) \frac{1}{x}
  6. 6

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

    f′′(x)=−(sin⁡(x)ln⁡(x))+cos⁡(x)1x+cos⁡(x)1x+sin⁡(x)−1x2f''(x) = -\left(\sin\left(x\right) \ln\left(x\right)\right) + \cos\left(x\right) \frac{1}{x} + \cos\left(x\right) \frac{1}{x} + \sin\left(x\right) \frac{-1}{x^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\sin\left(x\right) \ln\left(x\right)\right] = \cos\left(x\right) \ln\left(x\right) + \sin\left(x\right) \frac{1}{x} \]
\[ f(x) = \sin\left(x\right) \ln\left(x\right) \]
\[ u = \sin\left(x\right),\quad v = \ln\left(x\right) \]
\[ u' = \cos\left(x\right),\quad v' = \frac{1}{x} \]
\[ f'(x) = \left(\cos\left(x\right)\right)\left(\ln\left(x\right)\right) + \left(\sin\left(x\right)\right)\left(\frac{1}{x}\right) \]
\[ f'(x) = \cos\left(x\right) \ln\left(x\right) + \sin\left(x\right) \frac{1}{x} \]
\[ f''(x) = -\left(\sin\left(x\right) \ln\left(x\right)\right) + \cos\left(x\right) \frac{1}{x} + \cos\left(x\right) \frac{1}{x} + \sin\left(x\right) \frac{-1}{x^{2}} \]

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

xf(x)f'(x)
0.5-0.332312460.3505572
100.84147098
20.630276950.16619771
30.15503618-1.0405779

Questions

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

The derivative of sin⁡(x)ln⁡(x)\sin\left(x\right) \ln\left(x\right) is cos⁡(x)ln⁡(x)+sin⁡(x)1x\cos\left(x\right) \ln\left(x\right) + \sin\left(x\right) \frac{1}{x}.

What is the second derivative of sin(x) * ln(x)?

-(sin(x) * ln(x)) + cos(x) * 1 / x + cos(x) * 1 / x + sin(x) * -1 / x2x^{2}.

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

The product rule.

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

0.84147098.

Related derivatives

    Derivative of sin(x) * ln(x): cos(x) * ln(x) + sin(x) * 1 / x (with steps) | WordGPT