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Magazine Chapter 4: Problem 43
Find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. \(y=\left(1+\left(3 x^{2}-1\right)^{3}\right)^{2}\)
Short Answer
Expert verified
The derivative is \( \frac{d y}{d x} = 2(1 + (3x^2 - 1)^3) \cdot 18x((3x^2 - 1)^2) \).
Step by step solution
01
Understand the Problem
We need to differentiate the function \( y = \left(1 + \left(3x^2 - 1\right)^3\right)^2 \) with respect to \( x \) using the chain rule.
02
Apply the Outer Chain Rule
Identify the outer function as \( u^2 \) where \( u = 1 + \left(3x^2 - 1\right)^3 \). Differentiate the outer function with respect to \( u \): \( \frac{d}{du}(u^2) = 2u \).
03
Differentiate the Inner Function
Now differentiate the inner function \( u = 1 + (3x^2 - 1)^3 \) with respect to \( x \). First, identify an innermost function \( v = 3x^2 - 1 \) such that the inner function becomes \( u = 1 + v^3 \). Then, differentiate \( u \) with respect to \( v \): \( \frac{du}{dv} = 3v^2 \).
04
Differentiate the Innermost Function
Now differentiate the innermost function \( v = 3x^2 - 1 \). The derivative \( \frac{dv}{dx} = 6x \).
05
Chain the Derivatives Together
Using the chain rule repeatedly, we have that:\[\frac{d}{dx}y = \frac{d}{du}(u^2) \cdot \frac{du}{dv}(1 + v^3) \cdot \frac{dv}{dx}(3x^2 - 1) \, .\]Substitute the previously found derivatives:\[\frac{d}{dx}y = 2(1 + (3x^2 - 1)^3) \cdot 3((3x^2 - 1)^2) \cdot 6x \, .\]
06
Simplify the Derived Expression
We multiply all the parts together to get the full derivative:\[\frac{d y}{d x} = 2(1 + (3x^2 - 1)^3) \cdot 18x((3x^2 - 1)^2) \, .\]Simplify this expression further if needed for specific context or numbers, but this provides the derivative using the chain rule effectively.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Derivative
In calculus, the concept of a derivative is fundamental to understanding how functions change. The derivative represents the rate at which a quantity changes with respect to another. For a function \( y = f(x) \), the derivative \( \frac{dy}{dx} \) indicates the slope of the tangent line to the function at any point \( x \). It's like asking how steep the hill is for every position along a road.
For example, if you have \( y = x^2 \), the derivative tells you how the value of \( y \) increases as \( x \) increases. The derivative of \( y = x^2 \) is \( \frac{dy}{dx} = 2x \), which means at every point \( x \), the steepness or rate of change of \( y \) is \( 2x \).
For example, if you have \( y = x^2 \), the derivative tells you how the value of \( y \) increases as \( x \) increases. The derivative of \( y = x^2 \) is \( \frac{dy}{dx} = 2x \), which means at every point \( x \), the steepness or rate of change of \( y \) is \( 2x \).
- This helps us understand the behavior of functions: increasing, decreasing, finding minimum and maximum points, and even identifying points of inflection where the curve changes direction.
- Derivatives can also reveal acceleration or velocity in physical situations, making the understanding of derivatives necessary for fields like physics and engineering.
Chain Rule
The chain rule is a crucial method in calculus used to differentiate composite functions. A composite function is essentially a function within another function, like \( y = \left(1 + (3x^2 - 1)^3\right)^2 \). Here, different functions are layered, and to differentiate them, we apply the chain rule.
The chain rule states that to differentiate a composite function \( f(g(x)) \), you multiply the derivative of the outer function \( f \) by the derivative of the inner function \( g \). In math speak, if \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).
The chain rule states that to differentiate a composite function \( f(g(x)) \), you multiply the derivative of the outer function \( f \) by the derivative of the inner function \( g \). In math speak, if \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).
- In our exercise, the outer function is \( u^2 \) and the inner function is \( u = 1 + (3x^2 - 1)^3 \).
- We differentiate \( u^2 \) with respect to \( u \) getting \( \frac{d}{du}(u^2) = 2u \).
- Then, \( u = 1 + (3x^2 - 1)^3 \), so we need to differentiate this part, using the chain rule once more.
- This step-by-step chaining of derivatives is what gives the chain rule its name and allows us to handle complex compositions.
Differentiation Steps
Differentiation using the chain rule is a multi-step process where each step builds upon the last. Here is a simplified breakdown based on our example:
This chain of steps turns what might seem an overwhelming task into a manageable, methodical process, thus enabling the differentiation of very complex functions.
- **Step 1:** Identify the outer function and differentiate it. In the exercise, the outer function was \( (u)^2 \).
- **Step 2:** Identify the next inner function and differentiate it. Here, \( u = 1 + (3x^2 - 1)^3 \) was treated by finding \( \frac{du}{dv} = 3v^2 \), where \( v = 3x^2 - 1 \).
- **Step 3:** Continue until the innermost function is reached. For \( v = 3x^2 - 1 \), the derivative \( \frac{dv}{dx} = 6x \) is calculated.
- **Step 4:** Chain the derivatives using multiplication to assemble the full derivative. This involves combining \( \frac{d}{du}(u^2) \), \( \frac{du}{dv} \), and \( \frac{dv}{dx} \).
- **Step 5:** Simplify the resulting expression, making it easier to interpret or apply depending on the given context.
This chain of steps turns what might seem an overwhelming task into a manageable, methodical process, thus enabling the differentiation of very complex functions.
