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Magazine Chapter 2: Problem 6
When \(y=a x^{2} ; y^{1}=2 a x\); when \(y=a x^{3}\), then you should have found in Exercise 4 that \(y^{\prime}=3 a x^{2}\). Now suppose that \(y=a x^{4}\). What would you expect \(y^{\prime}\) to be? Verify your conjecture by applying the method of increments to \(y=a x^{4}\). Ans. \(y^{\prime}=4 a x^{3}\).
Short Answer
Expert verified
\(y^{\prime} = 4 a x^{3}\)
Step by step solution
01
State the Objective
We need to conjecture what the derivative of the function \(y = ax^{4}\) would be, based on the previous patterns observed, and then verify this by using the method of increments (which is another term for differentiation).
02
Identify the Pattern
We can see a pattern emerging from the given examples: \(y = ax^{2}\) leads to \(y^{\prime} = 2ax\), and \(y = ax^{3}\) leads to \(y^{\prime} = 3ax^{2}\). Observing that the exponent of \(x\) decreases by 1 and the coefficient becomes the original exponent times the constant \(a\), we can conjecture that for \(y = ax^{4}\), the derivative \(y^{\prime}\) will be \(4ax^{3}\).
03
Apply the Method of Increments
By definition, the method of increments (derivative) of \(y = ax^{4}\) can be found by using the limit definition of the derivative: \(y^{\prime} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}\) where \(\Delta y = a(x + \Delta x)^{4} - ax^{4}\).
04
Calculate the Increment \(\Delta y\)
Expanding \(a(x + \Delta x)^{4}\) and subtracting \(ax^{4}\) from it, we get the increment \(\Delta y = a[(x^4 + 4x^3\Delta x + 6x^2(\Delta x)^2 + 4x(\Delta x)^3 + (\Delta x)^4) - x^4]\). Simplifying this, we get \(\Delta y = a[4x^3\Delta x + 6x^2(\Delta x)^2 + 4x(\Delta x)^3 + (\Delta x)^4]\).
05
Compute the Derivative
To calculate \(y^{\prime}\) we divide \(\Delta y\) by \(\Delta x\) and take the limit as \(\Delta x\) approaches 0: \(y^{\prime} = \lim_{\Delta x \to 0} \frac{a[4x^3\Delta x + 6x^2(\Delta x)^2 + 4x(\Delta x)^3 + (\Delta x)^4]}{\Delta x}\). This simplifies to \(y^{\prime} = \lim_{\Delta x \to 0} [4ax^3 + 6ax^2\Delta x + 4ax(\Delta x)^2 + a(\Delta x)^3]\). As \(\Delta x\) approaches 0, all terms with \(\Delta x\) go to zero and we are left with \(y^{\prime} = 4ax^3\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Method of Increments
Understanding Calculus can be simplified by grasping core concepts like the method of increments, which is tied closely to the limit definition of the derivative. When considering a function such as y = ax4, the method of increments looks at how small changes in the input (x) affect the changes in the output (y).
To visualize it, imagine nudging the variable 'x' a tiny bit and seeing how 'y' wiggles in response. The ratio of the change in 'y' to the change in 'x' (called Δy and Δx, respectively) underpins this concept. As we let Δx shrink closer and closer to zero, the ratio evolves into the derivative of the function at that point. This process outlines, in essence, the method of increments used to decipher the nature of change within a function.
To visualize it, imagine nudging the variable 'x' a tiny bit and seeing how 'y' wiggles in response. The ratio of the change in 'y' to the change in 'x' (called Δy and Δx, respectively) underpins this concept. As we let Δx shrink closer and closer to zero, the ratio evolves into the derivative of the function at that point. This process outlines, in essence, the method of increments used to decipher the nature of change within a function.
Limit Definition of the Derivative
The limit definition of the derivative can be a bit tricky at first, but it's a powerful foundation of calculus. It formally defines what we mean by 'instantaneous rate of change,' akin to finding the exact speed of a car at a precise moment. For a function like y = ax4, the derivative at a point is the limit of the rate of change of 'y' with respect to 'x' as that rate of change is calculated over shorter and shorter intervals around the point.
The formula y' = \(\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}\) might look daunting, but it's essentially asking, 'What happens as the changes in 'x' get infinitely small?' By investigating this, we nail down the exact slope of the curve at a singular point, providing a snapshot of the function's behavior at that very instance.
The formula y' = \(\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}\) might look daunting, but it's essentially asking, 'What happens as the changes in 'x' get infinitely small?' By investigating this, we nail down the exact slope of the curve at a singular point, providing a snapshot of the function's behavior at that very instance.
Power Rule for Differentiation
The power rule is like a shortcut in the world of calculus, specifically when taking derivatives of polynomial functions. If you have a term like xn, where n is any real number, the power rule tells us that the derivative of that term is nxn-1. It simplifies the process we've seen with the limit definition by cutting straight to the answer. For instance, when applying this rule to y = ax4, we predict that the derivative, y', is 4ax3.
Why does this work? It's almost like peeling away layers 鈥� each differentiation peels off a layer, represented by the exponent, and the number of layers removed multiplies the original function (hence the coefficient being n). This method is quick, slick, and gets to the point without the full process of taking the limit.
Why does this work? It's almost like peeling away layers 鈥� each differentiation peels off a layer, represented by the exponent, and the number of layers removed multiplies the original function (hence the coefficient being n). This method is quick, slick, and gets to the point without the full process of taking the limit.
Derivative of Polynomial Functions
Turning to the derivative of polynomial functions, it's akin to understanding the growth patterns of various plants in a garden. Each term within a polynomial can be seen as a different species, each with its distinct rate of growth. In calculus, taking the derivative of a polynomial like y = ax4 gives us a new function that represents the rate at which the original function changes.
This rate of change is crucial: it tells us how 'steep' our function is at any given point, or how rapidly the 'y' value is changing as 'x' varies. For polynomial functions, which are sums of power functions, we apply the power rule to each term to get the overall rate of change. In essence, we're compiling a complete picture of change from the individual rates of each of the 'species' (or terms) in our polynomial 'garden'.
This rate of change is crucial: it tells us how 'steep' our function is at any given point, or how rapidly the 'y' value is changing as 'x' varies. For polynomial functions, which are sums of power functions, we apply the power rule to each term to get the overall rate of change. In essence, we're compiling a complete picture of change from the individual rates of each of the 'species' (or terms) in our polynomial 'garden'.
