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Chapter 2: Problem 28

Differentiate. $$ q(x)=\frac{4 x^{3}+2 x^{2}-5 x+13}{3} $$

Short Answer

Expert verified
The derivative is \( 4x^2 + \frac{4}{3}x - \frac{5}{3} \).

Step by step solution

01

Identify the function

The function to differentiate is given by \[ q(x) = \frac{4x^3 + 2x^2 - 5x + 13}{3} \]
02

Apply the constant multiple rule

Rewrite the function by factoring out the constant 1/3:\[ q(x) = \frac{1}{3} (4x^3 + 2x^2 - 5x + 13) \].
03

Differentiate the polynomial

Differentiate each term inside the parentheses separately. Remember,\[ \frac{d}{dx}(ax^n) = anx^{n-1} \].Applying this rule:\[ \frac{d}{dx} (4x^3) = 4 \cdot 3x^2 = 12x^2 \]\[ \frac{d}{dx} (2x^2) = 2 \cdot 2x = 4x \]\[ \frac{d}{dx} (-5x) = -5 \]\[ \frac{d}{dx} (13) = 0 \].
04

Combine the results and apply the constant

Sum up the differentiated terms and multiply by the constant \( \frac{1}{3} \):\[ \frac{dq}{dx} = \frac{1}{3} (12x^2 + 4x - 5) \].
05

Final simplified expression

Simplify the differentiated function:\[ \frac{dq}{dx} = 4x^2 + \frac{4}{3}x - \frac{5}{3} \].

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Key Concepts

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

constant multiple rule
In differentiation, the constant multiple rule is essential for simplifying the process of differentiating a function. This rule states that if you have a constant multiplied by a function, you can take the constant outside the differentiation process. For example, given a function \(cf(x)\), the differentiation is \(c \frac{d}{dx} f(x)\).

In our exercise, \(q(x) = \frac{1}{3}(4x^3 + 2x^2 - 5x + 13)\) uses the constant multiple rule. We can factor out \( \frac{1}{3} \) to simplify the process. This allows us to differentiate the polynomial inside the parentheses first and then multiply the result by \( \frac{1}{3} \).
power rule of differentiation
One of the fundamental rules in differentiation is the power rule. This rule is crucial for differentiating terms with exponents. For a term \(ax^n\), where \(a\) is a constant and n is a power, the differentiation is given by \(\frac{d}{dx}(ax^n) = anx^{n-1} \).

For example, in the problem:

  • \( \frac{d}{dx} (4x^3) = 4 * 3x^2 = 12x^2 \)

  • \( \frac{d}{dx} (2x^2) = 2 * 2x = 4x \)

  • \( \frac{d}{dx} (-5x) = -5 \)

  • \( \frac{d}{dx} (13) = 0 \)
By applying the power rule to each term, we effectively differentiate the entire expression.
simplifying expressions
After finding the derivatives of each term within the function, the next step is to simplify the expression. Simplification helps make the final result clearer and more manageable.

In the given exercise, after differentiating, we obtained:
\(\frac{d}{dx} = \frac{1}{3} (12x^2 + 4x - 5) \). The final step in differentiation involves multiplying back the constant \( \frac{1}{3} \) and simplifying:
\(\frac{dq}{dx} = 4x^2 + \frac{4}{3} x - \frac{5}{3} \).

Simplification often involves:
  • Combining like terms where possible

  • Reducing fractions

  • Ensuring that the expression is presented in the simplest form

This final simplification provides the most concise representation of the derivative function.

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Most popular questions from this chapter

Differentiate. $$ f(x)=\frac{x^{-1}}{x+x^{-1}} $$

The position of a pendulum that is slowing due to friction is given by $$s(t)=\frac{3 \cos t}{\sqrt{t}+1}$$ where \(t\) is time measured in seconds and the position \(s(t)\) is measured in inches. a) Find the velocity function for the pendulum. b) Find the velocity of the pendulum at time \(t=1\). c) Find the velocity of the pendulum at time \(t=\frac{\pi}{3}\).

Differentiate. $$ y=\frac{\tan t}{1+\sec t} $$

The dosage [or carboplatin chemotherapy drugs depends on several parameters of the drug as well as the age, weight, and sex of the patient. For a female patient, the formulas giving the dosage for a certain drug are $$D=0.85 A(c+25)$$ and $$c=\frac{(140-y) w}{72 x},$$ where \(A\) and \(x\) depend on which drug is used, \(D\) is the dosage in milligrams \((\mathrm{mg}), c\) is called the creatine clearance, \(y\) is the patient's age in years, and \(w\) is the patient's weight in \(\mathrm{kg} .{ }^{11}\) a) Suppose a patient is a 45 -yr-old woman and the drug has parameters \(A=5\) and \(x=0.6 .\) Use this information to find formulas for \(D\) and \(c\) that give \(D\) as a function of \(c\) and \(c\) as a function of \(w\). b) Use your formulas in part (a) to compute \(\frac{d D}{d c}\). c) Use your formulas in part (a) to compute \(\frac{d c}{d w}\). d) Compute \(\frac{d D}{d w}\). e) Interpret the meaning of the derivative \(\frac{d D}{d w}\).

Find an equation of the tangent line to the graph o \(y=\frac{\sqrt{x}}{x+1}\) at the point where \(x=1\).
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