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Chapter 7: Problem 15

Differentiate. \(y=\left(x^{2}-2 x+2\right) e^{x}\).

Short Answer

Expert verified
The derivative of the given function is: \(\frac{dy}{dx} = e^x(x^2 - 4x)\).

Step by step solution

01

Find the derivative of u(x)

To differentiate \(u(x) = x^2 - 2x + 2\) with respect to x, we find the derivative for each term: \[ \frac{d}{dx}(x^2) = 2x, \\ \frac{d}{dx}(-2x) = -2, \\ \frac{ d}{dx}(2) = 0 \] Summing these up gives: \[ u'(x) = 2x - 2 \]
02

Find the derivative of v(x)

To differentiate \(v(x) = e^x\) with respect to x, we use the fact that the derivative of \(e^x\) is itself: \[ v'(x) = e^x \]
03

Apply the product rule

Now that we have the derivatives of both \(u\) and \(v\), we can apply the product rule: \[ \frac{dy}{dx} = u'(x) \cdot v(x) + u(x) \cdot v'(x) \]
04

Substitute values and simplify

Substitute the expressions we found in Steps 1 and 2, along with the original functions for \(u(x)\) and \(v(x)\). This gives: \[ \frac{dy}{dx} = (2x - 2) \cdot e^x + (x^2 - 2x + 2) \cdot e^x \] Now, factor out the common factor of \(e^x\): \[ \frac{dy}{dx} = e^x (2x - 2 + x^2 - 2x + 2) \] Finally, simplify the expression inside the parentheses: \[ \frac{dy}{dx} = e^x(x^2 - 4x) \] Thus, the derivative of the given function is: \[ \frac{dy}{dx} = e^x(x^2 - 4x) \]

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Key Concepts

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

Differentiation
Differentiation is the process of finding the derivative of a function, which is a core concept in calculus. It essentially measures how a function changes as its input changes, formally described as the rate of change or slope at any given point on a curve.
To perform differentiation, we use rules like the power rule, product rule, and chain rule. For any function given by a polynomial, exponential, or other forms, these rules help us retireve its derivative.
  • The Power Rule is given by: if you have a function of the form \(f(x) = x^n\), the derivative is \(f'(x) = n \cdot x^{n-1}\).
  • The Product Rule is useful for determining the derivative of a product of two functions, as we'll discuss further.
  • The Chain Rule helps in differentiating composite functions, which are functions of functions.
When applied thoughtfully, these principles enable us to solve a wide range of differential calculus problems, providing essential tools in analyzing physical systems, optimizing functions, and more.
Derivative of Exponential Functions
Exponential functions, those with a base of e, such as \(e^x\), are unique and practical in calculus due to their special property: their rate of change equals themselves.
This is why when differentiating an exponential function:
  • The derivative of \(e^x\) is simply \(e^x\).
  • For a general exponential function \(e^{ax}\), where \(a\) is a constant, the derivative would be \(a \cdot e^{ax}\).
This property makes exponential functions vital in modeling growth and decay processes, such as population dynamics, radioactive decay, and financial calculations.
Polynomial Differentiation
Polynomial differentiation involves taking the derivative of functions expressed as a sum of terms, each consisting of a variable raised to a power and multiplied by a coefficient. These functions are particularly straightforward to differentiate using the aforementioned power rule.
Take, for example, the function given by \(u(x) = x^2 - 2x + 2\). Each term is differentiated individually:
  • For \(x^2\), apply the power rule to get \(2x\).
  • The term \(-2x\) differentiates to \(-2\).
  • The constant term, \(2\), has a derivative of 0 since it doesn鈥檛 change with \(x\).
The combination of these individual derivatives provides the complete polynomial derivative, allowing us to analyze and describe the behavior of polynomial functions over real numbers, making them essential in engineering and physics computations.

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