differentiate y = ln[sin^2((pi) x^(1/2)) + cos^2((pi)x)]

[Cronan52]

Find the Derivative:

y= ln(sin^2(3.14159(x^1/2))+(cos^2(3.14159x))

 

[skeeter]

please verify that the given function is ...

\(\displaystyle y = ln[sin^2(\pi \sqrt{x}) + cos^2(\pi x)]\)

 

[Cronan52]

that is correct

 

[skeeter]

\(\displaystyle \frac{d}{dx} ln(u) = \frac{u'}{u}\)

in this case, your "u" is \(\displaystyle sin^2(\pi \sqrt{x}) + cos^2(\pi x)\)

just a few applications of the chain rule to find u' ...

\(\displaystyle u' = 2sin(\pi \sqrt{x})cos(\pi \sqrt{x}) \frac{\pi}{2\sqrt{x}} - 2cos(\pi x)sin(\pi x)\pi\)

\(\displaystyle u' = \frac{\pi}{\sqrt{x}}sin(\pi \sqrt{x})cos(\pi \sqrt{x}) - 2\pi cos(\pi x)sin(\pi x)\)

you can finish up with \(\displaystyle \frac{u'}{u}\)