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Derivative of sin⁡(x)ln⁡(x)\frac{\sin\left(x\right)}{\ln\left(x\right)}

The derivative of sin⁡(x)ln⁡(x)\frac{\sin\left(x\right)}{\ln\left(x\right)} is cos⁡(x)ln⁡(x)−sin⁡(x)1xln⁡(x)2\frac{\cos\left(x\right) \ln\left(x\right) - \sin\left(x\right) \frac{1}{x}}{\ln\left(x\right)^{2}}.

f(x) =
ddx[sin⁡(x)ln⁡(x)]=cos⁡(x)ln⁡(x)−sin⁡(x)1xln⁡(x)2\frac{d}{dx}\left[\frac{\sin\left(x\right)}{\ln\left(x\right)}\right] = \frac{\cos\left(x\right) \ln\left(x\right) - \sin\left(x\right) \frac{1}{x}}{\ln\left(x\right)^{2}}

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Graph of f(x) = sin(x) / ln(x) and its derivative-3-2-1123-30-20-1010slope -3.262f(x)f'(x)tangent at x = 0.5
Graph of f(x) = sin(x) / ln(x) and its derivative. The dashed line is the tangent at x = 0.5; its slope is f'(0.5) = -3.2618.

How to work it out

  1. 1

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

    f(x)=sin⁡(x)ln⁡(x)f(x) = \frac{\sin\left(x\right)}{\ln\left(x\right)}
  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=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)(ln⁡(x))2f'(x) = \frac{\left(\cos\left(x\right)\right)\left(\ln\left(x\right)\right) - \left(\sin\left(x\right)\right)\left(\frac{1}{x}\right)}{\left(\ln\left(x\right)\right)^{2}}
  5. 5

    Simplify:

    f′(x)=cos⁡(x)ln⁡(x)−sin⁡(x)1xln⁡(x)2f'(x) = \frac{\cos\left(x\right) \ln\left(x\right) - \sin\left(x\right) \frac{1}{x}}{\ln\left(x\right)^{2}}
  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)−1x2))ln⁡(x)2−(cos⁡(x)ln⁡(x)−sin⁡(x)1x)2ln⁡(x)1x(ln⁡(x)2)2f''(x) = \frac{\left(-\left(\sin\left(x\right) \ln\left(x\right)\right) + \cos\left(x\right) \frac{1}{x} - \left(\cos\left(x\right) \frac{1}{x} + \sin\left(x\right) \frac{-1}{x^{2}}\right)\right) \ln\left(x\right)^{2} - \left(\cos\left(x\right) \ln\left(x\right) - \sin\left(x\right) \frac{1}{x}\right) 2 \ln\left(x\right) \frac{1}{x}}{\left(\ln\left(x\right)^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{\sin\left(x\right)}{\ln\left(x\right)}\right] = \frac{\cos\left(x\right) \ln\left(x\right) - \sin\left(x\right) \frac{1}{x}}{\ln\left(x\right)^{2}} \]
\[ f(x) = \frac{\sin\left(x\right)}{\ln\left(x\right)} \]
\[ u = \sin\left(x\right),\quad v = \ln\left(x\right) \]
\[ u' = \cos\left(x\right),\quad v' = \frac{1}{x} \]
\[ f'(x) = \frac{\left(\cos\left(x\right)\right)\left(\ln\left(x\right)\right) - \left(\sin\left(x\right)\right)\left(\frac{1}{x}\right)}{\left(\ln\left(x\right)\right)^{2}} \]
\[ f'(x) = \frac{\cos\left(x\right) \ln\left(x\right) - \sin\left(x\right) \frac{1}{x}}{\ln\left(x\right)^{2}} \]
\[ f''(x) = \frac{\left(-\left(\sin\left(x\right) \ln\left(x\right)\right) + \cos\left(x\right) \frac{1}{x} - \left(\cos\left(x\right) \frac{1}{x} + \sin\left(x\right) \frac{-1}{x^{2}}\right)\right) \ln\left(x\right)^{2} - \left(\cos\left(x\right) \ln\left(x\right) - \sin\left(x\right) \frac{1}{x}\right) 2 \ln\left(x\right) \frac{1}{x}}{\left(\ln\left(x\right)^{2}\right)^{2}} \]

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

xf(x)f'(x)
0.5-0.69166485-3.2618069
21.3118389-1.5466647
30.12845297-0.94010431

Questions

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

The derivative of sin⁡(x)ln⁡(x)\frac{\sin\left(x\right)}{\ln\left(x\right)} is cos⁡(x)ln⁡(x)−sin⁡(x)1xln⁡(x)2\frac{\cos\left(x\right) \ln\left(x\right) - \sin\left(x\right) \frac{1}{x}}{\ln\left(x\right)^{2}}.

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

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

The quotient rule.

What is the slope of sin(x) / ln(x) at x = 0.5?

-3.2618069.

Related derivatives

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