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Chapter 4: Problem 28

1-44. Find the derivative of each function. $$ f(x)=x^{2} e^{x}-2 \ln x+\left(x^{2}+1\right)^{3} $$

Short Answer

Expert verified
The derivative is \(2x e^x + x^2 e^x - \frac{2}{x} + 6x(x^2 + 1)^2\).

Step by step solution

01

Differentiate the First Term

The first term of the function is \( x^2 e^x \). Use the product rule to differentiate this term. The product rule states that \((uv)' = u'v + uv'\) for two functions \(u\) and \(v\). Here, let \(u = x^2\) and \(v = e^x\). Then \(u' = 2x\) and \(v' = e^x\). Thus, the derivative is \(2x e^x + x^2 e^x\).
02

Differentiate the Second Term

The second term is \(-2 \ln x\). Use the derivative of the natural logarithm, \((\ln x)' = \frac{1}{x}\). Therefore, the derivative of the term is \(-2 \cdot \frac{1}{x} = -\frac{2}{x}\).
03

Differentiate the Third Term

The third term is \((x^2 + 1)^3\). Use the chain rule, which states \((f(g(x)))' = f'(g(x))g'(x)\). Let \(u = x^2 + 1\) so \(f(u) = u^3\). Therefore, \(f'(u) = 3u^2\) and \(u' = 2x\). The derivative of the term is \(3(x^2 + 1)^2 \cdot 2x = 6x(x^2 + 1)^2\).
04

Combine All the Derivatives

Add up all the derivatives calculated from Steps 1 to 3 to form the derivative of the entire function. The overall derivative is \(2x e^x + x^2 e^x - \frac{2}{x} + 6x(x^2 + 1)^2\).

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Key Concepts

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

Product Rule
The Product Rule is a fundamental concept in differential calculus. It is essential when differentiating expressions where two functions are multiplied together. The rule is written as
  • If you have a function that is the product of two functions, say \(u(x)\) and \(v(x)\), the derivative is given by
  • \((uv)' = u'v + uv'\) .
This means you differentiate the first function, multiply by the second function as is, then add the product of the first function and the derivative of the second one.

In the problem, consider the term \(x^2 e^x\). Let \(u = x^2\) and \(v = e^x\). Therefore, \(u' = 2x\) and \(v' = e^x\). Applying the product rule gives us:
  • \((x^2)' \cdot e^x + x^2 \cdot (e^x)'\)
  • Which simplifies to \(2x e^x + x^2 e^x\).
By utilizing the product rule, you can simplify differentiating composite functions that include multiplicative components.
Chain Rule
The Chain Rule is a method for finding the derivative of composite functions. Composite functions are functions that are nested within another function. The Chain Rule states:
  • If \(y = f(g(x))\), then the derivative \(y'\) is given by
  • \((f(g(x)))' = f'(g(x)) \cdot g'(x)\) .
This means you differentiate the outer function, then multiply by the derivative of the inner function.

In the exercise, the term \((x^2 + 1)^3\) is a great example to demonstrate the Chain Rule. Here, set the inner function \(u = x^2 + 1\) and thus the outer function \(f(u) = u^3\).
  • The derivative of the outer function, \(f'(u) = 3u^2\).
  • The derivative of the inner function, \(u' = 2x\).
  • Therefore, using the Chain Rule, the derivative is \(3(x^2 + 1)^2 \cdot 2x = 6x(x^2 + 1)^2\).
    • By applying the Chain Rule, solving derivatives of complex functions with nested layers becomes systematic and manageable.
    Natural Logarithm Derivative
    The derivative of the natural logarithm function is a specific rule worth noting.
    The derivative of \(\ln x\) with respect to \(x\) is:
    • \((\ln x)' = \frac{1}{x}\)
    This result is crucial when dealing with calculus problems involving logarithmic terms. In this particular problem, the term \(-2 \ln x\) requires evaluating its derivative.
    To find this, apply the derivative rule:
    • The derivative of \(-2 \ln x\) is given as \(-2 \cdot \frac{1}{x} = -\frac{2}{x}\).
    Understanding the derivative of the natural logarithm helps simplify calculations in derivatives when logarithmic expressions are involved, allowing you to solve more complex equations efficiently.

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    Most popular questions from this chapter

    1-44. Find the derivative of each function. $$ f(x)=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}} $$

    \(61-62 .\) By calculating the first few derivatives, find a formula for the \(n\) th derivative of each function \((k\) is a constant \()\). $$ f(x)=e^{-k x} $$

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