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Chapter 4: Problem 1

Differentiate the functions given in Problems 1-22 with respect to the independent variable. $$ f(x)=4 x^{3}-7 x+1 $$

Short Answer

Expert verified
The derivative of the function is \(f'(x) = 12x^2 - 7\).

Step by step solution

01

Identify the Differentiation Rule

The function provided is a polynomial, specifically a cubic function, \[f(x) = 4x^3 - 7x + 1.\]To differentiate polynomials, use the power rule, which states that the derivative of \(ax^n\) is \(anx^{n-1}\).
02

Differentiate Each Term

Apply the power rule to each term separately:- The derivative of \(4x^3\) is \(3 \times 4 x^{3-1} = 12x^2\).- The derivative of \(-7x\) is \(-7 \times 1 x^{1-1} = -7\).- The derivative of the constant \(1\) is \(0\).
03

Write the Differentiated Function

Combine the differentiated results:\[f'(x) = 12x^2 - 7.\]

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Key Concepts

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

Understanding the Power Rule
The power rule is a fundamental concept in calculus that simplifies the differentiation of polynomial functions. When dealing with a term like \( ax^n \), the power rule states that its derivative is \( anx^{n-1} \). This rule is incredibly useful as it provides a quick way to find the derivative of terms with variables raised to a power.
For example, let's take the term \( 4x^3 \) from our original function. Here, \( a \) equals 4 and \( n \) equals 3.
  • According to the power rule, bring down the exponent 3 in front of the term.
  • Multiply the result, 3, by the coefficient 4 to get 12.
  • Subtract one from the exponent, changing \( x^3 \) to \( x^2 \).
Thus, the derivative of \( 4x^3 \) becomes \( 12x^2 \). The power rule makes differentiation faster and is crucial for handling more complex expressions efficiently.
Cubic Functions Explained
Cubic functions, like \( f(x) = 4x^3 - 7x + 1 \), are polynomial expressions where the highest degree of the variable is three. These functions generally have an \( x^3 \) term as their leading term. Understanding the shape and behavior of cubic functions can be important in calculus and graphing.
The graph of a cubic function typically:
  • Has an "S" shaped curve, featuring two bends or points of inflection.
  • Can extend from negative to positive infinity or vice versa.
  • Might intersect the x-axis at up to three points based on its roots.
For calculus, knowing that you're working with a cubic function helps when predicting the nature of its derivative. Differentiating reduces its degree by one, turning it into a quadratic, which can tell us more about the slope and concavity of the function at various points.
Polynomial Differentiation Steps
Differentiating a polynomial involves breaking the function down into individual terms and applying the differentiation rules, step-by-step. Polynomials consist of multiple terms, each involving variables raised to a power. This process is simplified by applying the power rule individually to each term.
Follow these steps to differentiate a polynomial:
  • Identify each term in the polynomial and the power of its variable.
  • Apply the power rule to each term. If a term is constant, its derivative is simply zero.
  • Combine all the differentiated terms to form the derivative of the complete function.
For the function \( f(x) = 4x^3 - 7x + 1 \), we:
  • Applied the power rule to \( 4x^3 \) to get \( 12x^2 \).
  • Did the same for \( -7x \), resulting in \( -7 \).
  • Took the derivative of the constant 1 as zero.
  • Combined them to find that \( f'(x) = 12x^2 - 7 \).
Polynomial differentiation is foundational in calculus and offers clear insight into the behavior and characteristics of functions.

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Most popular questions from this chapter

Use the formal definition to find the derivative of \(f(x)=\frac{1}{x+1}\) at \(x=0\).

Use the formal definition to find the derivative of \(y=\sqrt{x}\) at \(x=2\)

A measurement error in \(x\) affects the accuracy of the value \(f(x) .\) In each case, determine an interval of the form $$[f(x)-\Delta f, f(x)+\Delta f]$$ that reflects the measurement error \(\Delta x .\) In each problem, the quantities given are \(f(x)\) and \(x=\) true value of \(x \pm|\Delta x| .\) . \(f(x)=2 x, x=1 \pm 0.1\)

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ . \(f(x)=e^{2 x+1}\) at \(a=-1 / 2\)

Find the derivatives of the following functions: $$ f(x)=\cos ^{2}\left(2 x^{2}+3\right) $$
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