Question

differentiate $y = e^{x}-x^{2}arctan x$.

Explanation:

Step1: Differentiate \(e^x\)

The derivative of \(e^x\) with respect to \(x\) is \(e^x\).

Step2: Differentiate \(-x^2\arctan x\)

Use the product rule \((uv)^\prime = u^\prime v+uv^\prime\), where \(u = -x^2\), \(v=\arctan x\).

• \(u^\prime=-2x\)
• \(v^\prime=\frac{1}{1 + x^2}\)

Then \((-x^2\arctan x)^\prime=-2x\arctan x-\frac{x^2}{1 + x^2}\)

Step3: Combine the derivatives

\(y^\prime=(e^x)^\prime+(-x^2\arctan x)^\prime\)
\(y^\prime=e^x-2x\arctan x-\frac{x^2}{1 + x^2}\)

Answer:

\(y^\prime=e^x-2x\arctan x-\frac{x^2}{1 + x^2}\)