Problem 406 - Calculus Volume 3

Question

For the following exercises, find all second partial derivatives. \(g(t, x)=3 t^{2}-\sin (x+t)\)

Step-by-Step Solution

Answer

The second partial derivatives are: \( g_{tt} = 6 + \sin(t+x) \), \( g_{tx} = g_{xt} = \sin(t+x) \), \( g_{xx} = \sin(t+x) \).

Step 1: Calculate the First Partial Derivative with respect to t

To find the first partial derivative of the function \( g(t, x) = 3t^2 - \sin(t+x) \) with respect to \( t \), treat \( x \) as a constant. The derivative of \( 3t^2 \) with respect to \( t \) is \( 6t \).The derivative of \( -\sin(t+x) \) with respect to \( t \) is \( -\cos(t+x) \) because of the chain rule (\( \sin(t+x) \) differentiates to \( \cos(t+x) \), and the derivative of \( t+x \) with respect to \( t \) is \( 1 \)).So, the first partial derivative with respect to \( t \) is:\[ g_t(t,x) = 6t - \cos(t+x) \]

Step 2: Calculate the First Partial Derivative with respect to x

Next, find the derivative of the function \( g(t, x) = 3t^2 - \sin(t+x) \) with respect to \( x \), treating \( t \) as a constant.Since \( 3t^2 \) does not depend on \( x \), its derivative is zero.The derivative of \( -\sin(t+x) \) with respect to \( x \) is \( -\cos(t+x) \) due to the chain rule (since \( \sin(t+x) \) differentiates to \( \cos(t+x) \), and the derivative of \( t+x \) with respect to \( x \) is \( 1 \)).Thus, the first partial derivative with respect to \( x \) is:\[ g_x(t,x) = -\cos(t+x) \]

Step 3: Calculate the Second Partial Derivative with respect to t (g_tt)

Now compute the derivative of the first partial derivative \( g_t(t,x) = 6t - \cos(t+x) \) with respect to \( t \).The derivative of \( 6t \) with respect to \( t \) is \( 6 \).The derivative of \( -\cos(t+x) \) with respect to \( t \) is \( \sin(t+x) \) because \( \cos(t+x) \) differentiates to \( -\sin(t+x) \), and therefore the negative multiplication results in \( \sin(t+x) \).So, the second partial derivative with respect to \( t \) is:\[ g_{tt}(t,x) = 6 + \sin(t+x) \]

Step 4: Calculate the Mixed Partial Derivative g_tx

To find the mixed partial derivative, differentiate \( g_t(t,x) = 6t - \cos(t+x) \) with respect to \( x \).Since \( 6t \) does not depend on \( x \), its derivative is zero.The derivative of \( -\cos(t+x) \) with respect to \( x \) is \( \sin(t+x) \) by the same logic as before, where \( \cos(t+x) \) differentiates to \( -\sin(t+x) \). Thus, the mixed partial derivative \( g_{tx} \) is:\[ g_{tx}(t,x) = \sin(t+x) \]

Step 5: Calculate the Mixed Partial Derivative g_xt

Now, take the derivative of \( g_x(t,x) = -\cos(t+x) \) with respect to \( t \).The derivative of \( -\cos(t+x) \) with respect to \( t \) is \( \sin(t+x) \), for the same reasons previously outlined.Thus, the mixed partial derivative \( g_{xt} \) is:\[ g_{xt}(t,x) = \sin(t+x) \]

Step 6: Calculate the Second Partial Derivative with respect to x (g_xx)

Lastly, compute the derivative of\( g_x(t,x) = -\cos(t+x) \) with respect to \( x \).The derivative of \( -\cos(t+x) \) with respect to \( x \) is \( \sin(t+x) \).So, the second partial derivative with respect to \( x \) is:\[ g_{xx}(t,x) = \sin(t+x) \]

Key Concepts

Understanding Partial Differentiation

In mathematics, partial differentiation is the process of finding the derivative of a function with respect to one variable while holding the other variables constant. This is particularly useful in multivariable calculus where functions depend on more than one variable. For example, in the function \( g(t, x) = 3t^2 - \sin(t+x) \), we have two variables, \( t \) and \( x \). When calculating the partial derivative of \( g \) with respect to \( t \), we consider \( x \) to be constant, and only focus on how \( g \) changes as \( t \) changes.

• The partial derivative with respect to \( t \) is denoted as \( g_t(t, x) \).
• Similarly, when we differentiate with respect to \( x \), we treat \( t \) as a constant.
• The result is denoted as \( g_x(t, x) \).

Understanding partial derivatives is essential for solving many problems in engineering and physics, where functions often depend on several variables, and we want to see the sensitivity of the function with respect to one variable at a time.

Applying the Chain Rule in Partial Differentiation

The chain rule is a fundamental tool in calculus used to differentiate composite functions. In partial differentiation, the chain rule helps when differentiating expressions like \( \sin(t+x) \). Here's how it works:

• When you differentiate \( \sin(t+x) \) with respect to \( t \), the chain rule states you first differentiate the outer function \( \sin \) to get \( \cos(t+x) \).
• Next, multiply by the derivative of the inner function \( t+x \) with respect to \( t \), which is \( 1 \).

This gives us \( -\cos(t+x) \) as part of the derivative in Step 1 of the solution. Similarly, when differentiating \( \sin(t+x) \) with respect to \( x \), the process is the same because the inner derivative of \( t+x \) with respect to \( x \) is still \( 1 \). Consequently, the same steps and logic apply to find \( g_x(t, x) \) in Step 2. Understanding the chain rule ensures accuracy when working with derivatives of composite functions!

Exploring Mixed Partial Derivatives

Mixed partial derivatives refer to the derivatives of a function with respect to two or more different variables. For the function \( g(t, x) = 3t^2 - \sin(t+x) \), the first derivatives \( g_t \) and \( g_x \) are calculated separately, and then mixed partials are determined by further differentiating those results.

• \( g_{tx} \): First, differentiate \( g_t(t, x) \) with respect to \( x \), treating \( t \) as a constant.
• \( g_{xt} \): Similarly, differentiate \( g_x(t, x) \) with respect to \( t \), treating \( x \) as constant.

In our exercise, we find that \( g_{tx}(t,x) \) and \( g_{xt}(t,x) \) both equal \( \sin(t+x) \). This equality illustrates Clairaut's theorem on the equality of mixed partials, under conditions where the function is continuous and well-behaved. Mixed partial derivatives are crucial in fields like physics, where they can describe the behavior of a system under combined influences of multiple factors.

Comprehending Second Derivatives

Second derivatives provide insights into the curvature and concavity of a function. In the context of partial derivatives, second derivatives are derived by differentiating the first partials further with respect to their respective variables.

• \( g_{tt}(t, x) \): Differentiating \( g_t(t, x) = 6t - \cos(t+x) \) with respect to \( t \) gives this second derivative.
• \( g_{xx}(t, x) \): Derived by differentiating \( g_x(t, x) = -\cos(t+x) \) with respect to \( x \).

In the solution, \( g_{tt}(t, x) = 6 + \sin(t+x) \) and \( g_{xx}(t, x) = \sin(t+x) \), indicating the changing rates beyond the first derivative. Second derivatives help determine points of inflection and the nature of local extrema in multi-variable functions. They are indispensable in fields like optimization and economics, where understanding the shape of a function relative to its variables can inform critical decision-making processes.