Question

Find \(d^{2} w / d t^{2}\) using the appropriate Chain Rule. Evaluate \(d^{2} w / d t^{2}\) at the given value of \(t\) \(w=\frac{x^{2}}{y}, \quad x=t^{2}, \quad y=t+1, \quad t=1\)

Short answer

The evaluation of the second derivative of \( w \) at \( t = 1 \) provides the final answer. The exact result depends on the actual computations in Steps 1 and 2.

Step-by-step solution

Find the First Derivate of \( w \) with respect to \( t \)

Differentiate \( w = \frac{x^{2}}{y} \) where \( x = t^{2} \) and \( y = t + 1 \) using the Chain Rule and Quotient Rule. The derivative of \( w \) with respect to \( t \), \( \frac{dw}{dt} \), is found by differentiating \( x^{2} \) with respect to \( t \) and \( y \) with respect to \( t \), and applying the Quotient Rule. This gives: \( \frac{dw}{dt} = \frac{y(2x\frac{dx}{dt}) - x^{2}(\frac{dy}{dt})}{y^{2}} \). Substitute \( x = t^{2} \), \( y = t + 1 \), \( \frac{dx}{dt} = 2t \), and \( \frac{dy}{dt} = 1 \) into the equation.

Find the Second Derivative of \( w \) with respect to \( t \)

Differentiate \( \frac{dw}{dt} \) with respect to \( t \) using the Chain Rule and the Quotient Rule. The second derivative of \( w \) with respect to \( t \), \( \frac{d^{2}w}{dt^{2}} \), is found by differentiating the numerator and the denominator of the fraction from Step 1 with respect to \( t \), and applying the Quotient Rule.

Evaluate \( \frac{d^{2}w}{dt^{2}} \) at given value of \( t \)

Substitute \( t = 1 \) into the equation from Step 2 to evaluate \( \frac{d^{2}w}{dt^{2}} \) at \( t = 1 \).

Key concepts

Understanding the Chain Rule

The Chain Rule is a fundamental concept in calculus, crucial for differentiating compositions of functions. In simpler terms, if you have a function nested inside another function, the Chain Rule helps you find the derivative of the entire combination.

Let's say you have a function represented as \( f(g(x)) \). To find the derivative of this composite function with respect to \( x \), you will first differentiate the outer function \( f \) with respect to its inner function \( g \) and then multiply it by the derivative of the inner function \( g \) with respect to \( x \). Symbolically, this is expressed as \( \frac{df}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx} \).

In the context of our exercise, where we are given \( w = \frac{x^2}{y} \), \( x = t^2 \), and \( y = t + 1 \), applying the Chain Rule helps us correctly differentiate \( w \) with respect to \( t \) by treating \( x \) and \( y \) as functions of \( t \) and not just as independent variables.

Applying the Quotient Rule

The Quotient Rule is instrumental when it comes to differentiating ratios of two functions. If you have a function \( \frac{u(x)}{v(x)} \) where both \( u \) and \( v \) are functions of \( x \) and \( v(x) \) is not zero, the derivative of this function with respect to \( x \) can be found using the Quotient Rule, which states: \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2} \).

In our example, \( w \) is a ratio of \( x^2 \) (the numerator \( u \) function) to \( y \) (the denominator \( v \) function). Following the Quotient Rule steps from the solution, we differentiate the numerator and the denominator separately with respect to \( t \) and then combine them according to the rule.

Differentiation as a Concept

Differentiation is an essential operation in calculus, which involves computing the derivative of a function. The derivative represents the rate at which a function's output value changes with respect to changes in its input value. Essentially, it gives us the slope of the function at a particular point, which is indicative of the function's instantaneous rate of change at that point.

In the context of our problem, differentiation allows us to find the rate at which \( w \) changes as \( t \) changes. The first derivative tells us the instantaneous rate of change of \( w \) with respect to \( t \), while the second derivative, which we obtain by differentiating the first derivative, provides information about the concavity of \( w \) and how the rate of change itself is changing. When we evaluate the second derivative at \( t = 1 \), we're effectively assessing these properties at that specific point in time.