Question

In Exercises 15-28, find the derivative of the function. $$ y=x \arctan 2 x-\frac{1}{4} \ln \left(1+4 x^{2}\right) $$

Short answer

The derivative of the given function is \(y'=arctan(2x)\)

Step-by-step solution

Differentiate the product

The first part of the function is a product \(x*arctan(2x)\). The product rule states that the derivative of the product of two functions is the derivative of the first one times the second one plus the first one times the derivative of the second one. For our functions \(u=x\) and \(v=arctan(2x)\), we get \(u'=1\) and \(v'=\frac{2}{1+(2x)^2} = \frac{2}{1+4x^2}\) . This results in \(\frac{d}{dx}[x*arctan(2x)]=1*arctan(2x)+x*\frac{2}{1+4x^2}\)

Differentiate the logarithm function

The second part of the function is \(-\frac{1}{4}*ln(1+4x^2)\). The derivative of the log function \(ln(u)\) is \(\frac{u'}{u}\). Here \(u=1+4x^2\), so \(u'=8x\). This gives us the derivative \(\frac{-8x}{4*(1+4x^2)}\) which simplifies to \(-\frac{2x}{1+4x^2}\)

Combine both parts

The full function is the subtraction of the two parts differentiated in steps 1 and 2. Thus, the derivative of the function is \(\frac{d}{dx}[x*arctan(2x)-\frac{1}{4}*ln(1+4x^2)]=1*arctan(2x)+x*\frac{2}{1+4x^2}-\frac{2x}{1+4x^2}\)

Simplify the derivative

Simplify the derivative function: \(x*\frac{2}{1+4x^2}-\frac{2x}{1+4x^2}\) simplifies to 0. So, the final derivative function is \(y'=arctan(2x)\)