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

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

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

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Graph of f(x) = cos(x) * ln(x) and its derivative-3-2-1123-2-1123slope 0.5403f(x)f'(x)tangent at x = 1
Graph of f(x) = cos(x) * ln(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 0.5403.

How to work it out

  1. 1

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

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

    Product rule: (uv)′=u′v+uv′(uv)' = u'v + uv'. 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)f'(x) = \left(-\sin\left(x\right)\right)\left(\ln\left(x\right)\right) + \left(\cos\left(x\right)\right)\left(\frac{1}{x}\right)
  5. 5

    Simplify:

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

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

xf(x)f'(x)
0.5-0.608293882.0874776
100.54030231
2-0.28845101-0.83835037
3-1.0876179-0.48503367

Questions

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

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

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}.

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

The product rule.

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

0.54030231.

Related derivatives

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