Problem 50 - Calculus of a Single Variable

Question

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

Step-by-Step Solution

Answer

The derivative of the given function is \( \frac{7t}{4*(t^{2}+4)} \)

Step 1: Identify and differentiate the first part

The first part of the function is \(y = \ln (t^2+4)\). The derivative of the natural log function is \(1/x\), and here \(x\) is \(t^2+4\). So using the chain rule, which states that the derivative of \(h(g(x))\) is \(h'(g(x)) \cdot g'(x)\), the derivative of the first part would be \((1/(t^2+4)) \cdot (2t) = 2t/(t^2+4)\).

Step 2: Identify and differentiate the second part

The second part of the function is \(-1/2*\arctan(t/2)\). Here we use the rule that the derivative of \(arctan(u)\) is \(1/(1+u^2)\). Applying this rule, along with the chain rule as in Step 1, we find that the derivative of the second part is \(-1/2 * (1/(1+(t/2)^2)) * (1/2) = -t/(4*(t^2+4))\).

Step 3: Combine the results

By the rule that the derivative of a sum of functions is the sum of the derivatives, we add the derivatives from Step 1 and Step 2: \(2t/(t^2+4) - t/(4*(t^2+4)) = (8t - t) / (4*(t^2 + 4)) = 7t / (4*(t^2+4)).