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Chapter 2: Problem 43

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

Short Answer

Expert verified
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

01

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.
02

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 \( 2 \sin x + 2x \cos x \).
03

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 \( 2x e^{x} + x^{2} e^{x} \).
04

Combine the derivatives

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivatives
Understanding derivatives is key in calculus, especially when dealing with functions that change values. Derivatives allow us to determine the rate at which a function is changing at any given point.
Consider the function of a curve on a graph. The derivative tells you the slope of that curve at any particular point. This slope is what helps us understand whether the function is increasing, decreasing or staying constant.
For transcendental functions like the one in our exercise, derivatives might be slightly less straightforward because they could involve non-algebraic elements, like sin functions or exponential functions. But the core concept remains the same - finding how one quantity changes with respect to another.
Product Rule
The product rule is an essential tool when taking derivatives of functions that are products of two or more functions.
The product rule states that if you have two functions multiplied together, say \(f(x)\) and \(g(x)\), their derivative is given by:\[ (f \cdot g)' = f' \cdot g + f \cdot g' \] This helps simplify the process, ensuring that we don't inadvertently ignore critical components of the function.
**Example in Practice:** When we applied the product rule to the term \(2x \sin x\), we broke it down as \(u = 2x\) and \(v = \sin x\). We then followed the formula to get its derivative. The same process was necessary for the second term \(x^2 e^x\), underlining the rule's broad utility.
Differentiation
Differentiation is the process we use to find the derivative of a function. It involves systematically applying rules like the power rule, product rule, and chain rule to obtain the derivative.
For complex functions, differentiation can involve multiple steps and rules. In our exercise, we had to recognize that each term involved a product and used the product rule to differentiate them separately.
Key Steps in Differentiation:
  • Identify each component of the function you are differentiating.
  • Apply the appropriate differentiation rule.
  • Combine the derivatives of all components for the complete derivative of the function.
By systematically following these steps, even complex transcendental functions can be differentiated without error.

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