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Chapter 2: Problem 23

Compute the derivative of the given function. $$h(t)=7 t^{2}+6 t-2$$

Short Answer

Expert verified
The derivative of the function is \( h'(t) = 14t + 6 \).

Step by step solution

01

Identify the Rule

The function given is a polynomial in the form of \( h(t) = 7t^2 + 6t - 2 \). We will use basic differentiation rules: the derivative of \( t^n \) is \( nt^{n-1} \), and the derivative of a constant is zero.
02

Compute the Derivative of Each Term

Differentiate each term of the function separately:- The derivative of \( 7t^2 \) is \( 2 \times 7t^{2-1} = 14t \).- The derivative of \( 6t \) is \( 6 \times 1t^{1-1} = 6 \).- The derivative of \( -2 \), a constant, is 0.
03

Combine the Results

Combine the derivatives of each term to form the derivative of the given function:\[ h'(t) = 14t + 6 + 0 \]
04

Simplify the Derivative

Remove the unnecessary zero and rewrite the derivative:\[ h'(t) = 14t + 6 \]

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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 expression constructed from variables and coefficients using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. In our exercise, the function given is a polynomial: - The function is expressed as \( h(t) = 7t^2 + 6t - 2 \). Understanding polynomial functions is crucial for handling more complex equations. Here:- \(7t^2\) is a quadratic term.- \(6t\) represents the linear term.- \(-2\) is the constant term.Polynomial functions are smooth and continuous, making them easier to work with when applying differentiation rules.
Basic Differentiation Rules
Differentiation is a key process in calculus, involving finding the derivative of a function. For polynomial functions, basic differentiation rules allow us to determine how a function changes.The fundamental rules used include:
  • The derivative of a sum of functions is the sum of their derivatives.
  • The rule \(\frac{d}{dt}[t^n] = nt^{n-1}\) for finding derivatives of terms like \(t^n\).
  • The constant rule states that the derivative of a constant term is zero.
Understanding these rules simplifies the process of differentiating polynomial functions.
Constant Rule
The constant rule in differentiation states that the derivative of a constant is always zero. This is intuitive because constant terms do not change; they have no rate of change, which the derivative measures. For the original polynomial function, the term \(-2\) is a constant. Thus, according to the constant rule, its derivative becomes \(0\). It's simple yet essential to identify constant terms within polynomial equations for accurate differentiation.
Power Rule
The power rule is a pivotal component when differentiating polynomial functions. It provides a quick way to find the derivatives of terms where the variable is raised to a power.The power rule formula is \(\frac{d}{dt}[t^n] = nt^{n-1}\). Using the power rule:
  • The derivative of \(7t^2\) becomes \(14t\), as you multiply the exponent by the coefficient and reduce the exponent by one.
  • Similarly, \(6t\) is differentiated to \(6\), as \(t\) alone is considered \(t^1\), and applying the rule simplifies it accordingly.
This consistent approach simplifies differentiating polynomial terms efficiently.

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Most popular questions from this chapter

An implicitly defined function is given. Find \(\frac{d^{2} y}{d x^{2}} .\) Note: these are the same problems used in Exercises 13 through 16. $$\frac{x}{y}=10$$

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