Problem 7
Question
In Exercises \(7-14,\) find the differential \(d y\) of the given function. $$ y=3 x^{2}-4 $$
Step-by-Step Solution
Verified Answer
The differential \(dy\) of the function \(y=3x^2-4\) is \(dy = 6x \cdot dx\).
1Step 1: Write the function
The given function is \(y=3x^2-4\).
2Step 2: Differentiate the function
Taking the derivative of the function with respect to \(x\), yield \(dy/dx = 6x\). This was achieved by applying power rule which suggests that the derivative of \(x^n\) is \(nx^{n−1}\). Here \(n=2\) for \(3x^2\), thus the derivative is \(2*3x^{2-1} = 6x\). The constant term \(-4\) disappears during differentiation.
3Step 3: Write the differential
The differential of a function is given by \(dy = f'(x) dx\). As we've differentiated y earlier and found its derivative \(dy/dx=6x\), the expression for differential \(dy\) becomes \(dy = 6x \cdot dx\).
Key Concepts
DifferentiationPower RuleDerivative
Differentiation
Differentiation is a core concept in differential calculus, and it involves calculating the rate at which a function changes at any point on its curve. In simpler terms, it helps us find how a function's output varies with its input. Differentiation is performed with respect to a variable, typically denoted as "x," which may refer to time, distance, or any other independent variable.
The primary tool used in differentiation is the derivative, which can be thought of as the "slope" of a function at a given point. Differentiating a function involves several methods, but most often, it requires the application of standard rules like the product rule, quotient rule, chain rule, and the power rule.
The primary tool used in differentiation is the derivative, which can be thought of as the "slope" of a function at a given point. Differentiating a function involves several methods, but most often, it requires the application of standard rules like the product rule, quotient rule, chain rule, and the power rule.
- To apply differentiation effectively, one must understand these rules and recognize the types of functions involved.
- The process simplifies complex functions into their rate of change, making it invaluable for tasks like optimization and instantaneous rate analysis.
Power Rule
The power rule is one of the most fundamental tricks in calculus for finding derivatives. It simplifies the process of finding the derivative of a polynomial term that looks like \(x^n\). According to the power rule, the derivative of \(x^n\) is \(nx^{n-1}\).
This rule directly applies to any term where the variable \(x\) is raised to a power. Let's look at an example to make it clearer. Take \(y = 3x^2 - 4\), as seen in the exercise. Here, the term \(3x^2\) has the variable \(x\) with an exponent \(n=2\).
The power rule is widely applicable and extremely useful because many functions in calculus involve terms of the form \(x^n\). Understanding and applying this rule can greatly simplify finding derivatives of polynomial functions.
This rule directly applies to any term where the variable \(x\) is raised to a power. Let's look at an example to make it clearer. Take \(y = 3x^2 - 4\), as seen in the exercise. Here, the term \(3x^2\) has the variable \(x\) with an exponent \(n=2\).
- The derivative according to the power rule is \(2 \cdot 3x^{2-1}\), which simplifies to \(6x\).
- The constant term \ \(-4\) is differentiated to zero, due to the rule that the derivative of a constant is always zero.
The power rule is widely applicable and extremely useful because many functions in calculus involve terms of the form \(x^n\). Understanding and applying this rule can greatly simplify finding derivatives of polynomial functions.
Derivative
A derivative represents the rate of change of a function with respect to a variable. It is a fundamental aspect of calculus, providing crucial insights into the behavior of functions.
When we say we're taking the derivative of a function, we're interested in how that function changes as its input changes. Consider it like measuring the slope of a line, but for curves. For the given function in the exercise, \(y = 3x^2 - 4\), its derivative \(dy/dx\) is \(6x\). This means that at any point \(x\), the function is changing at a rate of \(6x\) times.
When we say we're taking the derivative of a function, we're interested in how that function changes as its input changes. Consider it like measuring the slope of a line, but for curves. For the given function in the exercise, \(y = 3x^2 - 4\), its derivative \(dy/dx\) is \(6x\). This means that at any point \(x\), the function is changing at a rate of \(6x\) times.
- The concept of the derivative is essential in understanding graphs, as it tells us whether a function is increasing or decreasing at any given point.
- Every function can have many derivatives, such as the first derivative, second derivative, and so on, each providing deeper insights into its behavior like concavity and inflection points.
- For practical applications, derivatives are used extensively in fields such as physics to find velocities and accelerations, in economics to determine marginal costs, and in many more domains.
Other exercises in this chapter
Problem 6
Find the two \(x\) -intercepts of the function \(f\) and show that \(f^{\prime}(x)=0\) at some point between the two \(x\) -intercepts. $$ f(x)=x(x-3) $$
View solution Problem 6
Identify the open intervals on which the function is increasing or decreasing. $$ f(x)=x^{4}-2 x^{2} $$
View solution Problem 7
Determine the open intervals on which the graph is concave upward or concave downward. \(y=2 x-\tan x, \quad\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)
View solution Problem 7
Numerical and Graphical Analysis In Exercises 3-8, use a graphing utility to complete the table and estimate the limit as \(x\) approaches infinity. Then use a
View solution