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Chapter 4: Problem 25

find \(\frac{d}{d x} f(x)\) $$ f(x)=1-x-x^{2}-x^{3}-10 \ln x $$

Short Answer

Expert verified
The derivative is \(-1 - 2x - 3x^2 - \frac{10}{x}\).

Step by step solution

01

Identify the function components

The function given is \( f(x) = 1 - x - x^2 - x^3 - 10 \ln x \). It consists of a constant, polynomial terms, and a logarithmic term.
02

Differentiate the constant and polynomial terms

Differentiate the constant and polynomial terms: the derivative of \( 1 \) is \( 0 \), the derivative of \( -x \) is \( -1 \), the derivative of \( -x^2 \) is \( -2x \), and the derivative of \( -x^3 \) is \( -3x^2 \).
03

Differentiate the logarithmic term

Differentiate \( -10 \ln x \). The derivative of \( \ln x \) is \( \frac{1}{x} \), so the derivative of \( -10 \ln x \) is \( -10 \times \frac{1}{x} = -\frac{10}{x} \).
04

Combine the derivatives

Combine all the derivative results: \( \frac{d}{dx}f(x) = 0 - 1 - 2x - 3x^2 - \frac{10}{x} \).
05

Write the final derivative expression

The final expression for the derivative is \( -1 - 2x - 3x^2 - \frac{10}{x} \).

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Key Concepts

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

Polynomial Derivative
Differentiating a polynomial is a fundamental skill in calculus. Polynomials are expressions made up of variables raised to whole-number exponents and are often combined through addition, subtraction, or multiplication by constants. The process of finding a derivative involves using the power rule.When applying the power rule, follow these steps:
  • Identify the terms in the polynomial. For example, in the expression \( -x - x^2 - x^3 \), there are three terms.
  • Apply the power rule: For any term \( ax^n \), the derivative is \( nax^{n-1} \). This results from dropping the exponent as a coefficient and reducing the exponent by one.
  • For our terms: \(-x\) becomes \(-1\), \(-x^2\) becomes \(-2x\), and \(-x^3\) becomes \(-3x^2\).
Constant terms like \(1\) completely disappear, as their derivatives are zero. Practice will enhance your efficiency in handling polynomial derivatives thoroughly.
Logarithmic Differentiation
Logarithmic differentiation is used when dealing with functions multiplying or dividing with variables in their powers, or in the function itself. It simplifies complex expressions, especially those involving products or quotients involving exponential or logarithmic terms.Key points to remember:
  • The derivative of \(\ln x\) is \(\frac{1}{x}\). When you have a constant multiplier like \(-10\) in \(-10 \ln x\), simply apply it to the derivative of \(\ln x\).
  • For \(-10 \ln x\), multiply \(-10\) by \(\frac{1}{x}\) to get \(-\frac{10}{x}\).
  • This technique reduces the complexity of dealing with logarithmic expressions within larger derivatives.
  • Apply it carefully to avoid errors, particularly in combining terms like power functions and logs.
Calculus Techniques
Calculus techniques include a wide variety of methods to solve complex mathematical problems, especially in finding derivatives or integrals. For instance, combining differentiation rules is often necessary for multi-component functions. Essentials of these techniques:
  • Approach a problem by first decomposing the function into manageable parts, such as identifying polynomial and logarithmic terms separately.
  • Differentiation rules can be applied stepwise, like the power rule for polynomials or logarithmic differentiation for log terms.
  • Combine derivative results carefully to avoid mistaking the results or missing terms.
      • This ensures precision in arriving at the final result.
    Practicing these techniques will improve your hands-on calculus, making intricate problems more approachable with systematic solutions.

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Most popular questions from this chapter

Biology Talekar and coworkers \(^{91}\) showed that the number \(y\) of the eggs of \(O\). furnacalis per mung bean plant was approximated by \(y=f(x)=-57.40+63.77 x-\) \(20.83 x^{2}+2.36 x^{3},\) where \(x\) is the age of the mung bean plant and \(2 \leq x \leq 7\). Find \(f^{\prime}(x),\) and explain what this means. Use the quadratic formula to show that \(f^{\prime}(x)>0\) for \(2 \leq x \leq 7\). What does this say about the preference of O. furnacalis to laying eggs on older plants?

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Biology The free water vapor diffusion coefficient \(D\) in soft woods \(^{21}\) is given by $$ D(T)=\frac{167.2}{P}\left(\frac{T}{273}\right)^{1.75} $$ where \(T\) is the temperature and \(P\) is the constant atmospheric pressure measured in appropriate units. Find the rate of change of \(D\) with respect to \(T\).

Economies of Scale In 1955 Surdis \(^{87}\) obtained records from a utility company regarding its trench digging operations. The records show that the unit cost \(C(n)\) per foot of earth removed by the mechanical trencher is given approximately by $$ C(n)=\frac{15.04+0.74 n}{25 n} $$ where \(n\) is the number of hours worked per day. a. Graph. Find values for \(C^{\prime}(n)\) at \(x=2,4,6,\) and \(8 .\) Interpret what these numbers mean. What is happening? Units costs for hand digging was found to be \(\$ 0.60 .\) b. Approximate the number of hours worked at which using the trench digging machinery is more cost effective than hand digging.
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