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Chapter 2: Problem 17

In Exercises 3鈥�24, use the rules of differentiation to find the derivative of the function. $$ s(t)=t^{3}+5 t^{2}-3 t+8 $$

Short Answer

Expert verified
\(s'(t) = 3t^2 + 10t - 3\).

Step by step solution

01

Differentiate the first term

Firstly, using the power rule, differentiate the term \(t^3\). The derivative is \(3t^(3-1) = 3t^2\).
02

Differentiate the second term

Then, continue to differentiate the term \(5t^2\). The derivative becomes \(5*2*t^(2-1) = 10t\).
03

Differentiate the third term

The next term is \(-3t\). By again applying the power rule, the derivative will be \(-3*1*t^(1-1) = -3\).
04

Differentiate the constant term

The final term \(8\) is a constant, and the derivative of a constant is zero.
05

Combine results

Combine all the differentiated terms to obtain the derivative: \(s'(t) = 3t^2 + 10t -3\).

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Key Concepts

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

Differentiation rules
Differentiation is the process of finding the derivative of a function. A derivative represents the rate at which a function is changing at any given point, offering insights into the behavior of the function. There are several rules used for differentiation, which help in simplifying the process across various types of functions.
These rules include:
  • The power rule
  • The constant rule
  • The constant multiple rule
  • The sum and difference rules
By applying these rules systematically, you can find the derivative of elementary and composite functions efficiently. It's important to understand and practice these rules as they form the foundation for solving more complex calculus problems later.
Power rule
The power rule is one of the most frequently used rules in differentiation. It provides a simple method for finding the derivative of terms in the form of a power function, which is represented as \(x^n\).
According to the power rule, if \(f(x) = x^n\), then the derivative \(f'(x)\) is given by:
\[ f'(x) = nx^{n-1} \] This rule essentially allows you to bring down the power as a coefficient and then reduce the power by one.
For example, to find the derivative of \(t^3\), you would:
  • Bring the 3 down as a coefficient, giving you \(3t\)
  • Subtract 1 from the power, making it \(3t^2\)
The power rule is straightforward but incredibly powerful in handling polynomial terms.
Constant function derivative
In differentiation, a constant function is a function that always returns the same value, regardless of the input. The simplest form can be expressed as \(f(x) = c\), where \(c\) is a constant.
The derivative of a constant function is always zero. This is because a constant value does not change, and the derivative, measuring how a function changes, indicates no change.
For instance, in our problem, the term 8 is a constant. The derivative of 8 is simply 0, reflecting this lack of change. Remembering this rule will save time when differentiating expressions with constant terms by eliminating unnecessary steps.
Polynomial function differentiation
Differentiating polynomial functions involves applying basic differentiation rules to each term individually. A polynomial is composed of terms like \(at^n\), each of which can be differentiated separately using the power rule.
For the function \(s(t)=t^{3}+5 t^{2}-3 t+8\), we differentiate:
  • \(t^3\) using the power rule gives \(3t^2\)
  • \(5t^2\) results in \(10t\), using the constant multiple rule along with the power rule
  • \(-3t\) becomes \(-3\), using the power rule
  • The constant \(8\) offers no contribution to the derivative, as its derivative is zero
Finally, we combine these results to find \(s'(t) = 3t^2 + 10t - 3\). Each term is treated individually, following the differentiation rules, and put together to give the complete differentiated polynomial function.

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Most popular questions from this chapter

Finding a Second Derivative In Exercises \(91-98\) , find the second derivative of the function. $$ f(x)=\frac{x}{x-1} $$

True or False? In Exercises \(93-96\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a function is differentiable at a point, then it is continuous at that point.

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Using Related Rates In Exercises \(1-4,\) assume that \(x\) and \(y\) are both differentiable functions of \(t\) and find the required values of \(d y / d t\) and \(d x / d t .\) $$ \begin{aligned} x^{2}+y^{2}=25 & \text { (a) } \frac{d y}{d t} \text { when } x=3, y=4 \quad \frac{d x}{d t}=8 \\ & \text { (b) } \frac{d x}{d t} \text { when } x=4, y=3 \quad \frac{d y}{d t}=-2 \end{aligned} $$

Differential Equations In Exercises \(125-128\) , verify that the function satisfies the differential equation. $$ \text{Function} \quad \text{Differential Equation} $$ $$ y=2 x^{3}-6 x+10 \quad-y^{\prime \prime \prime}-x y^{\prime \prime}-2 y^{\prime}=-24 x^{2} $$
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