Question

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ f(t)=-2 t^{2}+3 t-6 $$

Short answer

The derivative of the function \( -2t^2 + 3t - 6 \) is \( -4t + 3 \).

Step-by-step solution

Identify the Terms

In the function \(f(t) = -2t^2 + 3t - 6\), identify the 3 terms: \( -2t^2\), \(3t\), and \(-6\).

Apply the Power Rule

Power rule states that the derivative of \( t^n \) where 'n' is a real number, is \(n t^{n-1}\). Apply this rule to the terms \( -2t^2 \) and \( 3t \). For the term \( -2t^2 \), the derivative is \( -4t \) (taking \( -2 \) as a coefficient and \(2\) out front using the power rule). Likewise, for the term \( 3t \), the derivative is \( 3 \) (as \( t \) is essentially \( t^1 \), applying the power rule gives \(1 . 3t^{1-1} = 3\).)

Apply the Constant Rule

The constant rule states that the derivative of a constant term is zero. So, the derivative of the term \(-6\) is \(0\).

Combine the Derivatives

Combine the found derivatives of individual terms to form the derivative of the whole function. Hence, the derivative of \(f(t) = -2t^2 + 3t - 6\) is \(f'(t) = -4t + 3\).

Key concepts

Understanding the Power Rule

The power rule is a simple, yet powerful tool in calculus that makes finding derivatives easier. If you have a function term like \( t^n \), the power rule helps you find its derivative quickly. Here's how it works: the derivative of \( t^n \) is \( n \cdot t^{n-1} \). This rule tells you to multiply the exponent by the term and then subtract one from the exponent. For example, in the term \(-2t^2\), apply the power rule by multiplying \(2\) (the exponent) by \(-2\) (the coefficient), which gives \(-4\). Then, reduce the exponent by one to get \(-4t^{1}\) or just \(-4t\). The simplicity of the power rule makes it widely used and essential for solving calculus problems efficiently.

Applying the Constant Rule

The constant rule is another fundamental concept in calculus. It states that the derivative of a constant number is always zero. This rule simplifies calculations by eliminating terms when differentiating. For a function like \( -6 \), this means its derivative is simply \(0\).

The constant rule helps reduce clutter in your calculations and focuses your attention on terms that change. By applying this rule, you're acknowledging that constants don't affect the rate of change in a function. This is valuable in more complex calculations, ensuring you're focusing only on parts of the function that impact its behavior.

Exploring Derivatives

Derivatives are core to understanding change in mathematics. They provide insight into how a function behaves, offering information about the function's rate of change and direction at any given point. In general,

• The derivative gives us the slope of the tangent line to the function at a specific point.
• It helps to determine when a function is increasing or decreasing.

In our example, the function \( f(t) = -2t^2 + 3t - 6 \), the derivative, \( f'(t) = -4t + 3 \), shows the rate at which \( f(t) \) changes with respect to \( t \). By finding derivatives, you gain deeper insights into function behavior, aiding in problem-solving across a range of applications.

Grasping Functions

A function is essentially a relationship between inputs and outputs where each input is related to one output. Understanding functions is crucial in calculus since they form the basis for further study into their behavior through differentiation and integration.

• Functions describe patterns and relationships in mathematics and real-world phenomena.
• In our example, \( f(t) = -2t^2 + 3t - 6 \), \( f(t) \) offers a way to transform each \( t \) into a unique value.

Functions can represent anything from simple linear relationships to complex phenomena. Through calculus, you learn not only to describe these relationships but also to analyze how they change and interact over time or space.