Problem 37
Question
Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Other Symbols for the Derivative. $$\text { If } y=2 x^{3}-3, \text { find } y^{\prime}$$
Step-by-Step Solution
Verified Answer
\(y' = 6x^{2}\)
1Step 1: Identify the Function
Start by identifying the function that needs differentiation. In this case, the function given is \(y=2x^{3}-3\).
2Step 2: Apply the Power Rule
To find \(y'\), apply the power rule for differentiation which states that the derivative of \(x^{n}\) with respect to \(x\) is \(nx^{n-1}\). Thus, the derivative of \(2x^{3}\) with respect to \(x\) is \(3 \times 2x^{3-1}\) or \(6x^{2}\).
3Step 3: Differentiate the Constant
The derivative of a constant is zero. Therefore, the derivative of \(3\) with respect to \(x\) is 0.
4Step 4: Combine the Derivatives
Combine the derivatives of all terms in the function to get \(y'\). \(y' = 6x^{2} + 0\), which simplifies to \(y' = 6x^{2}\).
Key Concepts
DifferentiationPower RuleCalculus
Differentiation
Differentiation is a fundamental concept in calculus, a mathematical process used to find how a function changes at any given point. It is the procedure of calculating the derivative of a function. To understand differentiation, imagine you are tracking the position of a car over time. If you want to calculate the car's speed at any specific moment, you would use differentiation to find the rate at which the car's position changes with time. In mathematical terms, if you have a function such as \( y = f(x) \), the derivative of this function, denoted as \( f'(x) \), represents the rate at which the function's output \(y\) changes with respect to the input \(x\). By differentiating a function, we can determine things like velocity, if the function represents distance over time, or slope if it represents a graph's rise over run. It's an essential tool that allows us to predict and understand the behavior of various real-world phenomena.
Power Rule
The Power Rule is one of the most utilized rules in the process of differentiation, especially when dealing with polynomial functions. The Power Rule states that to differentiate a term of the form \( x^n \), where \(n\) is any real number, you can bring down the exponent \(n\) as a coefficient, multiply it by the existing coefficient, and decrease the exponent by one. In mathematical notation, the rule is expressed as \( \frac{d}{dx} x^n = nx^{n-1} \).
For example, applying it to the function \( 2x^3 \) from the exercise, we would multiply the exponent 3 by the coefficient 2 and subtract one from the exponent to obtain \( 6x^2 \). This rule simplifies the process of finding derivatives by providing a straightforward method to operate on each term of the polynomial independently. It's important to remember that the Power Rule applies to any term with an exponent, including negative exponents and fractional exponents, with the exception of the constant term.
For example, applying it to the function \( 2x^3 \) from the exercise, we would multiply the exponent 3 by the coefficient 2 and subtract one from the exponent to obtain \( 6x^2 \). This rule simplifies the process of finding derivatives by providing a straightforward method to operate on each term of the polynomial independently. It's important to remember that the Power Rule applies to any term with an exponent, including negative exponents and fractional exponents, with the exception of the constant term.
Calculus
Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It is divided into two major parts: differential calculus, which studies the rate of change of functions, and integral calculus, which looks at the accumulation of quantities and finding areas under curves. The concept of calculus is foundational in many fields including physics, engineering, economics, and even biology.
Calculus allows us to understand motion, growth, and change at a deep level. For instance, when an astronaut is launched into space, calculus helps determine the trajectory and fuel consumption. When an economist analyzes the changing market trends, they use calculus to predict future events. Even in the development of medical drugs, calculus assists in the understanding of the rates of reaction and diffusion. Learning calculus, one starts with understanding the basics of functions and limits, progresses to mastery of differentiation and integration, and often culminates with the study of more complex topics like differential equations and series. It is a vast and powerful mathematical tool that helps to describe the world around us.
Calculus allows us to understand motion, growth, and change at a deep level. For instance, when an astronaut is launched into space, calculus helps determine the trajectory and fuel consumption. When an economist analyzes the changing market trends, they use calculus to predict future events. Even in the development of medical drugs, calculus assists in the understanding of the rates of reaction and diffusion. Learning calculus, one starts with understanding the basics of functions and limits, progresses to mastery of differentiation and integration, and often culminates with the study of more complex topics like differential equations and series. It is a vast and powerful mathematical tool that helps to describe the world around us.
Other exercises in this chapter
Problem 36
When the Limit Is an Expression $$\lim _{d \rightarrow 0} \frac{\left[(x+d)^{2}+1\right]-\left(x^{2}+1\right)}{d}$$
View solution Problem 37
Write the differential \(d y\) in terms of \(x, y,\) and \(d x\) for each implicit relation. $$2 x^{2}+3 x y+4 y^{2}=20$$
View solution Problem 37
Find the derivative of each function.. $$y=\frac{x^{1 / 2}}{x^{1 / 2}+1}$$
View solution Problem 37
When the Limit Is an Expression $$\lim _{d \rightarrow 0} \frac{(x+d)^{3}-x^{3}}{d}$$
View solution