Question

Find the first partial derivatives of the following functions. $$F(u, v, w)=\frac{u}{v+w}$$

Short answer

Question: Find the first-order partial derivatives of the function F(u, v, w) = u / (v + w) with respect to u, v, and w. Answer: The first-order partial derivatives of the function F(u, v, w) are: 1. With respect to u: ∂F/∂u = 1 / (v + w) 2. With respect to v: ∂F/∂v = -u / (v + w)^2 3. With respect to w: ∂F/∂w = -u / (v + w)^2

Step-by-step solution

Find the partial derivative with respect to u

To find the partial derivative of F(u, v, w) with respect to u, we differentiate F with respect to u while holding v and w constant: $$\frac{\partial F}{\partial u} = \frac{\partial}{\partial u} \left(\frac{u}{v+w}\right)$$ Since v+w is a constant with respect to u, we can use the rule for the derivative of a constant times a function: $$\frac{\partial F}{\partial u} = \frac{1}{v+w}$$

Find the partial derivative with respect to v

To find the partial derivative of F(u, v, w) with respect to v, we differentiate F with respect to v while holding u and w constant: $$\frac{\partial F}{\partial v} = \frac{\partial}{\partial v} \left(\frac{u}{v+w}\right)$$ Since v+w is the denominator and u is a constant with respect to v, we can use the Quotient Rule: $$\frac{\partial F}{\partial v} = \frac{(v+w)(0)-u(1)}{(v+w)^2}$$ Simplify the expression: $$\frac{\partial F}{\partial v} = \frac{-u}{(v+w)^2}$$

Find the partial derivative with respect to w

To find the partial derivative of F(u, v, w) with respect to w, we differentiate F with respect to w while holding u and v constant: $$\frac{\partial F}{\partial w} = \frac{\partial}{\partial w} \left(\frac{u}{v+w}\right)$$ Since v+w is the denominator and u is a constant with respect to w, we can use the Quotient Rule again: $$\frac{\partial F}{\partial w} = \frac{(v+w)(0)-u(1)}{(v+w)^2}$$ Simplify the expression: $$\frac{\partial F}{\partial w} = \frac{-u}{(v+w)^2}$$ The first-order partial derivatives of the function F(u, v, w) are: $$\frac{\partial F}{\partial u} = \frac{1}{v+w}$$ $$\frac{\partial F}{\partial v} = \frac{-u}{(v+w)^2}$$ $$\frac{\partial F}{\partial w} = \frac{-u}{(v+w)^2}$$

Key concepts

Calculus

At the heart of calculus lies the study of how things change. It's a branch of mathematics focusing on derivatives, integrals, and limits. Derivatives are vital, as they represent the rate at which quantities change. In our exercise, we delve into a specific kind of derivative known as the partial derivative, crucial for understanding changes in multivariable functions.

Understanding calculus opens up a world of physics, engineering, economics, and more, since it equips us to model and predict natural phenomena with a mathematical language. So, when we compute the first partial derivatives of the function \( F(u, v, w)=\frac{u}{v+w} \), we're essentially exploring how the function changes as each variable is altered independently.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions with more than one variable. Unlike single-variable calculus, where we deal with functions in the form \( f(x) \), multivariable calculus tackles functions like our \( F(u, v, w) \) that depend on several variables.

It adds complexity and richness to the study since we can examine how a function changes in various directions - not just along the \( x \) or \( y \) axes but in an \( n \) dimensional space. The partial derivatives we calculate give us the rate of change in each individual direction, providing a multi-faceted view of our function's behavior.

Quotient Rule

The Quotient Rule is a technique in calculus for finding the derivative of a quotient of two functions. It's beautifully captured by the formula \( \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \).

When we apply it to partial derivatives, it helps us to differentiate ratios where each part may also be a function of multiple variables. In our exercise, the Quotient Rule was used to find the derivatives with respect to \( v \) and \( w \) since \( F(u, v, w)=\frac{u}{v+w} \) involves a numerator \( u \) and a denominator \( v+w \), both of which are functions of the variables.

Partial Derivative with Respect to u

A partial derivative with respect to \( u \) measures how a function changes as \( u \) changes, with all other variables held constant. In the context of our function \( F(u, v, w) \), calculating \( \frac{\partial F}{\partial u} \) tells us the rate at which \( F \) changes for small changes in \( u \), given that \( v \) and \( w \) remain fixed.

This calculation is simpler than it might seem since \( v+w \) behaves like a constant when differentiating with respect to \( u \) and is therefore excluded from the differentiation process, giving us \( \frac{\partial F}{\partial u} = \frac{1}{v+w} \) for our exercise.

Partial Derivative with Respect to v

The partial derivative \( \frac{\partial F}{\partial v} \) reveals how \( F \) reacts to infinitesimal movements in \( v \) while keeping \( u \) and \( w \) constant. When we take the partial derivative of our function \( F(u, v, w) \) with respect to \( v \) and apply the Quotient Rule, we take into account that \( u \) is independent of \( v \) and thus, we treat \( u \) as a constant.

Doing this yields the derivative \( \frac{\partial F}{\partial v} = \frac{-u}{(v+w)^2} \), providing insight into how the presence of \( v \) in the denominator influences the behavior of the function.

Partial Derivative with Respect to w

Similarly to the derivative with respect to \( v \) the partial derivative \( \frac{\partial F}{\partial w} \) tells us about the sensitivity of \( F \) to changes in \( w \) when \( u \) and \( v \) are held steady. Here, we again use the Quotient Rule, accentuating the effect of \( w \) in the \( v+w \) term in the denominator. The outcome \( \frac{\partial F}{\partial w} = \frac{-u}{(v+w)^2} \) mirrors that of the derivative with respect to \( v \) since it also lies in the denominator, which shows a particular symmetry in how \( F \) responds to changes in \( v \) and \( w \) individually.