Question

Use logarithmic differentiation to find the derivative of the function. y = (ln x)cos 2x

Answer 1

`y = \ln(x) \cos(2x)`

Take the logarithm of both sides.

`\ln(y) = \ln\bigg(\ln(x) \cos(2x)\bigg)`

Expand the right side.

`\ln(y) = \ln\bigg(\ln(x)\bigg) + \ln\bigg(\cos(2x)\bigg)`

Use the chain rule to differentiate both sides with respect to
`x`.

`\frac{y'}y = (\bigg(\ln(x)\bigg)')/(\ln(x)) + (\bigg(\cos(2x)\bigg)')/(\cos(2x))`

`\frac{y'}y = \frac1{x\ln(x)} - (\sin(2x)\bigg(2x\bigg)')/(\cos(2x))`

`\frac{y'}y = \frac1{x\ln(x)} - 2 \tan(2x)`

Solve for
`y'`.

`\frac{y'}y = (1 - 2x \ln(x) \tan(2x))/(x\ln(x))`

`y' = (1 - 2x \ln(x) \tan(2x))/(x\ln(x))\,y`

`y' = (1 - 2x \ln(x) \tan(2x))/(x\ln(x))\,\bigg(\ln(x) \cos(2x)\bigg)`

`y' = \boxed{\frac{\cos(2x) - 2x \ln(x) \sin(2x)}x}`