Problem 58 - Calculus of a Single Variable

Question

Find the derivative of the function. \(y=\arctan \frac{x}{2}-\frac{1}{2\left(x^{2}+4\right)}\)

Step-by-Step Solution

Answer

The derivative of \(y=\arctan \frac{x}{2}-\frac{1}{2\left(x^{2}+4\right)}\) is \(y' = \frac{1}{2+x^{2}}-\frac{x}{(x^{2}+4)^2}\).

Step 1. Apply the Chain Rule to arc tangent function

The derivative of \(y = \arctan(u)\) is \(y' = \frac{u'}{1+u^{2}}\) where \(u = \frac{x}{2}\) and \(u' = \frac{1}{2}\). Substituting these values gives: \(\frac{1}{2(1+(\frac{x}{2})^{2})}\)

Step 2. Simplify the expression

The simplified form is \(\frac{1}{2+x^{2}}\).

Step 3. Apply the Chain Rule to the fraction

The derivative of \(y = v^{-1}\) where \(-1\) is a power is \(y' = -v^{-2}v'\). Identify \(v = 2(x^{2}+4)\) and \(v' = 4x\). Substitute to get \(-\frac{4x}{4(x^{2}+4)^{2}}\).

Step 4. Simplify the expression

This simplifies to \(-\frac{x}{(x^{2}+4)^2}\).

Step 5. Add both parts

Finally, sum results from Step 2 and Step 4 which gives \(\frac{1}{2+x^{2}}-\frac{x}{(x^{2}+4)^2}\).