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

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

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

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Graph of f(x) = cos(x) / ln(x) and its derivative-3-2-1123-20-15-10-5510slope -2.961f(x)f'(x)tangent at x = 0.5
Graph of f(x) = cos(x) / ln(x) and its derivative. The dashed line is the tangent at x = 0.5; its slope is f'(0.5) = -2.9615.

How to work it out

  1. 1

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

    f(x)=cos⁡(x)ln⁡(x)f(x) = \frac{\cos\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=cos⁡(x)u = \cos\left(x\right) and v=ln⁡(x)v = \ln\left(x\right).

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

    Differentiate each piece:

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

    Put them in the rule:

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

    Simplify:

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

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

xf(x)f'(x)
0.5-1.266084-2.9614814
2-0.60037298-0.87876133
3-0.901130.14496166

Questions

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

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

What is the second derivative of cos(x) / ln(x)?

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

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

The quotient rule.

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

-2.9614814.

Related derivatives

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