Question

Find the derivative of the transcendental function. $$ y=2 x \sin x+x^{2} e^{x} $$

Short answer

The derivative of \( y = 2x \sin x + x^{2} e^{x} \) is \( y' = 2 \sin x + 2x \cos x + 2x e^{x} + x^{2} e^{x} \).

Step-by-step solution

Differentiate terms separately

First, treat each term separately. The function \( y \) has two terms: \( 2x \sin x \) and \( x^{2} e^{x} \). It's important to recognise these are product of two functions.

Apply Product Rule to First term

Next, apply the product rule to the first term \( 2x \sin x \). That is, let \( u = 2x \) and \( v = \sin x \). Then \( u' = 2 \) and \( v' = \cos x \). The derivative by the product rule is \( 2 \sin x + 2x \cos x \).

Apply Product Rule to Second Term

For the second term, \( x^{2} e^{x} \), let \( w = x^{2} \) and \( z = e^{x} \). Then \( w' = 2x \) and \( z' = e^{x} \). Hence, the derivative is \( 2x e^{x} + x^{2} e^{x} \).

Combine the derivatives

Combine the derivatives of the terms to get the derivative of the whole function: \( y' = 2 \sin x + 2x \cos x + 2x e^{x} + x^{2} e^{x} \).