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Chapter 2: Problem 14

Find the derivative of each function. $$f(x)=3 x^{2}-5 x+4$$

Short Answer

Expert verified
The derivative of the function is \( f'(x) = 6x - 5 \).

Step by step solution

01

Identify the Function Components

The given function is \( f(x) = 3x^2 - 5x + 4 \). We need to find the derivative with respect to \( x \). The function is a polynomial consisting of three terms: \( 3x^2 \), \( -5x \), and \( 4 \).
02

Apply the Power Rule to Each Term

Use the power rule for derivatives, which states that \( \frac{d}{dx}[x^n] = nx^{n-1} \). Apply this rule to each term in the function:- **For** \( 3x^2 \), the derivative is \( 2 \times 3x^{2-1} = 6x \).- **For** \( -5x \), the derivative is \( 5 \times 1x^{1-1} = -5 \).- **For** the constant \( 4 \), the derivative is \( 0 \), as the derivative of a constant is always zero.
03

Combine the Derivatives

Now, combine the derivatives of each term. Sum up the results from Step 2: The derivative of \( f(x) = 3x^2 - 5x + 4 \) is \( f'(x) = 6x - 5 \).

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Key Concepts

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

Polynomial Function
A polynomial function is an important type of mathematical expression that consists of variables raised to whole number powers, along with coefficients. It looks like this:
  • Each term is a product of a constant and a variable raised to a power.
  • Polynomial functions can have one or more terms.
  • They are typically written in the form of \(a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0\).
In the given exercise, the function \(f(x) = 3x^2 - 5x + 4\) is a polynomial.The components include the terms:
  • \(3x^2\) which is a quadratic term,
  • \(-5x\) which is a linear term, and
  • \(4\) which is a constant term.
Polynomial functions are smooth, continuous curves that can extend infinitely in both directions on a graph. Understanding these functions is vital for analyzing curves, maximum and minimum points and solving equations.
Power Rule
The power rule is a basic but crucial tool in calculus for differentiating functions. When we have an expression of the form \(x^n\), the power rule helps to find its derivative. The rule states: \[\frac{d}{dx}[x^n] = nx^{n-1}\]This means you multiply the original exponent by the coefficient of the term and then reduce the exponent by one.Let's see how this works on each term from our function:
  • For \(3x^2\), applying the power rule gives \(2 \times 3x^{2-1} = 6x\).
  • The term \(-5x\) is actually \(-5x^1\), so its derivative becomes \(1 \times -5x^{1-1} = -5\).
Constants are a special case. The exponent is zero for constants, so their derivative is always zero. This is why in the case of the constant \(4\) the derivative is \(0\).
Differentiation
Differentiation is the process of finding the derivative of a function, which measures the rate at which a function's value changes as its input changes.
  • It is a fundamental operation in calculus used to analyze and predict changes in quantities.
  • Derivatives help to find slopes of tangents, velocity in physics, and optimized solutions in economics.
In our example, the function \(f(x) = 3x^2 - 5x + 4\) is transformed by the differentiation process.We individually differentiate each term and then combine them:
  • We first apply the power rule to each variable term:
  • The term \(3x^2\) becomes \(6x\),
  • \(-5x\) becomes \(-5\), and
  • The constant \(4\) becomes \(0\).
Adding these results, the derivative of the given polynomial function is \(f'(x) = 6x - 5\).Differentiation reveals critical information about the function, such as how steeply it rises or falls, providing valuable insights into the behavior of the function over different intervals.

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Most popular questions from this chapter

Find an expression for the derivative of the composition of three functions, \(\frac{d}{d x} f(g(h(x)))\). [Hint: Use the Chain Rule twice.]

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$\sqrt{x^{2}-9}+5$$

Use the Generalized Power Rule to find the derivative of each function. $$h(z)=\left(5 z^{2}+3 z-1\right)^{3}$$

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$\left(x^{2}-x\right)^{-3}$$

Use the Generalized Power Rule to find the derivative of each function. $$f(x)=\left(\frac{x-1}{x+1}\right)^{5}$$
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