Question

$$ \text { If } w=5 y-2 x \text { state } \frac{\partial^{2} w}{\partial x^{2}} \text { and } \frac{\partial^{2} w}{\partial y^{2}} \text {. } $$

Short answer

Answer: The second-order partial derivatives are $$\frac{\partial^{2}w}{\partial x^{2}} = 0$$ and $$\frac{\partial^{2}w}{\partial y^{2}} = 0$$.

Step-by-step solution

Differentiate $$w$$ with respect to $$x$$ and $$y$$

First, we will find the first-order partial derivatives of $$w$$ with respect to $$x$$ and $$y$$. To do this, we need to differentiate the function $$w = 5y - 2x$$ partially with respect to each variable, treating the other variable as constant. Differentiating with respect to $$x$$: $$\frac{\partial w}{\partial x} = -2$$ Differentiating with respect to $$y$$: $$\frac{\partial w}{\partial y} = 5$$

Differentiate the first-order partial derivatives with respect to $$x$$ and $$y$$

Now that we have the first-order partial derivatives, we can find the second-order partial derivatives by differentiating these derivatives with respect to $$x$$ and $$y$$ again. Differentiating $$\frac{\partial w}{\partial x} = -2$$ with respect to $$x$$: $$\frac{\partial^{2}w}{\partial x^{2}} = 0$$ Differentiating $$\frac{\partial w}{\partial y} = 5$$ with respect to $$y$$: $$\frac{\partial^{2}w}{\partial y^{2}} = 0$$

Conclusion

The second-order partial derivatives of the function $$w = 5y - 2x$$ are: $$\frac{\partial^{2}w}{\partial x^{2}} = 0$$ and $$\frac{\partial^{2}w}{\partial y^{2}} = 0$$.

Key concepts

Partial Derivatives

In multivariable calculus, partial derivatives allow us to understand how a function changes as we vary one of its variables while keeping the others constant. When dealing with a function of two variables, such as \( w = 5y - 2x \), we are often interested in how the function changes with respect to each individual variable. This is essential for analyzing functions in fields like physics and engineering.

To find the partial derivative of \( w \) with respect to \( x \), we treat \( y \) as a constant and differentiate only with respect to \( x \). In the case of our function, this process yields \( \frac{\partial w}{\partial x} = -2 \). Similarly, when we differentiate \( w \) with respect to \( y \), treating \( x \) as constant, we obtain \( \frac{\partial w}{\partial y} = 5 \). These derivatives inform us of the rate and direction of change of \( w \) with respect to changes in \( x \) and \( y \).

• When the partial derivative is constant (like \(-2\) or \(5\)), it suggests a uniform rate of change.
• Partial derivatives are crucial for analyzing the slope or trend within multi-dimensional data.

Second-Order Derivatives

Once we have calculated the first-order partial derivatives, we can investigate the rate of change of these derivatives themselves by finding second-order derivatives. This involves taking the derivative of the derivative, and it provides deeper insight into the curvature and concavity of the original function.

In the exercise with the function \( w = 5y - 2x \), we are tasked with finding the second-order partial derivatives with respect to both \( x \) and \( y \). For \( \frac{\partial w}{\partial x} = -2 \), differentiating again with respect to \( x \) yields \( \frac{\partial^{2} w}{\partial x^{2}} = 0 \). Similarly, for \( \frac{\partial w}{\partial y} = 5 \), differentiating with respect to \( y \) gives us \( \frac{\partial^{2} w}{\partial y^{2}} = 0 \).

• Second-order partial derivatives indicate how a rate of change itself changes; a zero value here suggests linear behavior.
• They are used in optimization problems and in determining points of inflection.

Differentiation

Differentiation is a fundamental concept in calculus that measures how a function changes as its inputs shift. It is the foundation upon which partial and second-order derivatives are built. Differentiation can be performed on functions of a single variable or functions with multiple variables, as seen in our exercise.

For single-variable functions, differentiation gives us the slope of the tangent line at any point. For multi-variable functions, partial differentiation helps us find the slope concerning one variable while the others are constant. This technique extends to find derivatives of higher order, such as second-order derivatives, providing insights into the function's behavior beyond simple rates of change.

• Differentiation is essential for studying dynamics in a variety of fields.
• It aids in constructing mathematical models to simulate real-world scenarios.
• Both partial derivatives and second-order derivatives rely on differentiation principles.