Problem 228
Question
For the following exercises, find \(\frac{d y}{d x}\) for each function. $$y=\left(3 x^{2}+3 x-1\right)^{4}$$
Step-by-Step Solution
Verified Answer
\( \frac{dy}{dx} = 4(3x^2 + 3x - 1)^3 (6x + 3) \)
1Step 1: Identify the Rule to Use
Notice that the function given is a composite function, i.e., a function within a function. The outer function is something raised to the fourth power, and the inner function is a polynomial. To differentiate it, we will use the chain rule.
2Step 2: Differentiate the Outer Function
Let the inner function be denoted as \( u = 3x^2 + 3x - 1 \). The outer function can now be expressed as \( y = u^4 \). Differentiating the outer function with respect to \( u \), we have \( \frac{d y}{d u} = 4u^3 \).
3Step 3: Differentiate the Inner Function
Differentiate \( u = 3x^2 + 3x - 1 \) with respect to \( x \). The derivative is \( \frac{d u}{d x} = 6x + 3 \).
4Step 4: Apply the Chain Rule
According to the chain rule, \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \). Substitute in the derivatives found in Steps 2 and 3:\[ \frac{dy}{dx} = 4u^3 \times (6x + 3) \].
5Step 5: Substitute Back 'u'
Replace \( u \) back with the original expression \( 3x^2 + 3x - 1 \) to finalize the solution:\[ \frac{dy}{dx} = 4(3x^2 + 3x - 1)^3 \times (6x + 3) \].
Key Concepts
DifferentiationComposite FunctionPolynomial Function
Differentiation
Differentiation is a fundamental concept in calculus that deals with how functions change. It's like understanding the rate at which something happens. In our problem, differentiation helps us find how the function \( y = (3x^2 + 3x - 1)^4 \) changes as \( x \) changes.
To differentiate, we often apply rules like the product rule, quotient rule, and chain rule. These tools make it possible to tackle more complex functions effectively. When we differentiate a function, we symbolically represent this as \( \frac{dy}{dx} \). This denotes the derivative of \( y \) with respect to \( x \).
Differentiation is not just academic; it has real-world applications. It's used to find velocities in physics, optimize processes in economics, and even in machine learning for training models. Understanding how to differentiate, therefore, is important for not only math but various fields.
To differentiate, we often apply rules like the product rule, quotient rule, and chain rule. These tools make it possible to tackle more complex functions effectively. When we differentiate a function, we symbolically represent this as \( \frac{dy}{dx} \). This denotes the derivative of \( y \) with respect to \( x \).
Differentiation is not just academic; it has real-world applications. It's used to find velocities in physics, optimize processes in economics, and even in machine learning for training models. Understanding how to differentiate, therefore, is important for not only math but various fields.
Composite Function
A composite function arises when one function is nested within another. Think of it as a box inside a box where you have to open the outer one to get to the inner one. The given exercise is a great example of this concept with \( y = (3x^2 + 3x - 1)^4 \). This is called a composite because it's made up of the outer function \( u^4 \) and the inner function \( u = 3x^2 + 3x - 1 \).
To solve problems involving composite functions, we often use the chain rule. The chain rule helps us differentiate the outer function while considering the inner function's derivative. This intricate dance between layers of functions is the heart of the composite function.
Once you grasp the idea of composite functions, it opens up a whole new world of possibilities in calculus, allowing you to simplify complex expressions step-by-step.
To solve problems involving composite functions, we often use the chain rule. The chain rule helps us differentiate the outer function while considering the inner function's derivative. This intricate dance between layers of functions is the heart of the composite function.
Once you grasp the idea of composite functions, it opens up a whole new world of possibilities in calculus, allowing you to simplify complex expressions step-by-step.
Polynomial Function
Polynomial functions are expressions that involve powers of \( x \). In our exercise, the polynomial function is \( 3x^2 + 3x - 1 \). These functions are vital in mathematics because they are simple yet incredibly versatile. A polynomial of degree \( 2 \) is called a quadratic function, and it's defined by its terms, each consisting of a coefficient, the variable \( x \), and a power.
The special thing about polynomial functions is their smooth curves, which makes them easy to work with. They have known roots and turning points, which aid in solving various real-world problems.
Differentiating a polynomial is straightforward because we simply bring down the power and reduce it by one, as shown when differentiating \( 3x^2 + 3x - 1 \) to obtain \( 6x + 3 \). This process simplifies many calculus problems, making it an essential tool for calculus learners.
The special thing about polynomial functions is their smooth curves, which makes them easy to work with. They have known roots and turning points, which aid in solving various real-world problems.
- Linear polynomials involve powers of \( x \) no higher than 1.
- Quadratic polynomials have \( x \) raised to the power of 2.
- Cubic polynomials go up to \( x^3 \).
Differentiating a polynomial is straightforward because we simply bring down the power and reduce it by one, as shown when differentiating \( 3x^2 + 3x - 1 \) to obtain \( 6x + 3 \). This process simplifies many calculus problems, making it an essential tool for calculus learners.
Other exercises in this chapter
Problem 227
For each of the following exercises, a. decompose each function in the form \(y=f(u)\) and \(u=g(x),\) and b. find \(\frac{d y}{d x}\) as a function of \(x .\)
View solution Problem 227
For the following exercise, a. decompose each function in the form \(y=f(u)\) and \(u=g(x),\) and b. find \(\frac{d y}{d x}\) as a function of \(x\). $$ y=-6 \s
View solution Problem 228
Find \(\frac{d y}{d x}\) for each function. $$ y=\left(3 x^{2}+3 x-1\right)^{4} $$
View solution Problem 229
For the following exercises, find \(\frac{d y}{d x}\) for each function. $$y=(5-2 x)^{-2}$$
View solution